Harmonic transmission characteristics for ultra-long distance AC transmission lines based on frequency-length factor

Electric Power Systems Research 182 (2020) 106189

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Harmonic transmission characteristics for ultra-long distance AC transmission lines based on frequency-length factor☆

T

Chang Chena, Honggeng Yanga, Weikang Wangb, Mirka Mandichb, Wenxuan Yaob,c,*, Yilu Liub,c a

College of Electrical Engineering, Sichuan University, Chengdu, 610065 Sichuan, China Department of Electrical Engineering and Computer Science, The University of Tennessee, Knoxville, 37996 TN, United States c Oak Ridge National Laboratory, Oak Ridge, 37830 TN, United States b

A R T I C LE I N FO

A B S T R A C T

Keywords: Frequency-length factor Half-wavelength transmission line Harmonic transmission characteristics Ultra-long-distance transmission line

Distributed parameter characteristics of Ultra-long Distance Transmission Lines (UDTLs) are prone to cause harmonic amplification. Amplified harmonic components deteriorate the quality of energy delivery, and consequently affect the safety and operations of a power grid. This paper proposes a method based on FrequencyLength Factor (FLF) to investigate the harmonic transmission characteristics (HTCs) for UDTLs, including HalfWavelength Transmission Lines (HWTLs). The proposed method considers the impact of line loss and reveals the comprehensive effects of line length, operation mode, and frequency on HTCs analysis for UDTLs. It is proved that only inter-harmonics can be amplified and cause resonance in lossy standard HWTLs propagation. Using the proposed method, the severity of harmonic amplification is quantitatively calculated. Additionally, this paper provides a fast evaluation approach for potential resonance frequencies of UDTLs, mitigating power quality issues caused by harmonic amplification. The effectiveness of the proposed method is verified by PSCAD simulation.

1. Introduction Global electricity demand is forecasted to increase by nearly 50% by 2050, making the intelligent allocation of energy resources becomes more important than ever [1]. Numerous strategies have been proposed to optimize the allocation of energy, including the development of ultra-high voltage power systems, the utilization of renewable energies, and the global energy interconnection [2]. One such strategy is the installation of efficient long-distance power transmission technology [3]. This is of particular importance for many countries, such as Brazil, China, and Russia. Various Ultra-long Distance Transmission lines (UDTLs) over 500 km will be applied in these countries, because their load centers are located far away from power production facilities [4]. Particularly, the distance from some of the load centers to energy resources is close to half of a power frequency wavelength (about 3000 km and 2500 km for 50 Hz and 60 Hz systems, respectively) [5–7]. Since Half-Wavelength Transmission Line (HWTL) has good transmission stability performance and economic advantage, it is considered as a competitive alternative for ultra-long distance power transmission [8]. These facts indicate the increasing importance of UDTL analysis. Unfortunately, the increased penetration of renewable energy

sources has caused increasing levels of harmonic pollution in the grid [40]. Harmonic pollution is transferred and amplified in UDTLs due to their large capacitance. Because harmonics greatly deteriorate power quality, it is essential to understand the harmonic transmission characteristics (HTCs) of UDTLs. Several methods have been proposed for resonance analysis of UDTLs, which can be divided into five categories. (a). In Refs. [9–11], the distribution characteristics of harmonics along long lines are analyzed based on the frequency scan method, with a simplification approach of ignoring line loss. However, the amplitude and phase distribution of transmission coefficient are different in lossy lines. Therefore, the characteristics of lossless UDTLs derived by this simplified method cannot be directly applied to lossy lines. As a result, the HTCs of lossy UDTLs still need further analysis. (b). Ref. [12] reveals the resonance frequencies for high-voltage transmission system with long lines. Ref. [13] studies harmonic resonance in parallel grid-connected photovoltaic inverters. Both methods use the lumped-element model of transmission line. However, the distributed parameters characteristics of lines cannot be neglected when analyzing long lines, making this approach inapplicable to UTDLs.

☆ ⁎

This work was supported in part by the National Natural Science Foundation of China (No. 51877141). Corresponding author at: Department of Electrical Engineering and Computer Science, The University of Tennessee, Knoxville, 37996 TN, United States. E-mail address: [email protected] (W. Yao).

https://doi.org/10.1016/j.epsr.2019.106189 Received 17 July 2019; Received in revised form 22 November 2019; Accepted 29 December 2019 Available online 27 January 2020 0378-7796/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature

a b Tpq * TIu TUM TIM Zre Ure Kc L* Δνf , L*

Constants f0 V0 λ0 N Pn

Fundamental frequency Fundamental voltage Fundamental wavelength Integers Natural power

Variables ZL ZT SSC P f fr h Xsys Zsys Zc XCP Γ α β λ L νf , L

The real part of n The negative imaginary part of n Transfer functions The amplification criterion of current disturbance sources The local maximum values of |TUu | The local maximum values of |TIu | Reference impedance Reference voltage Ssc (short-circuit ratio) P Relative derivative length Derivative of FLF

Indices

Load impedance Transformer impedance Short circuit capacity Active power Frequency of disturbance source Resonance frequency Harmonic order fV02/(f0 SSC ) (reactance of the receiving system) jXsys (impedance of the receiving system) The characteristic impedance of a transmission line Equivalent reactance of shunt capacitors Propagation constant Attenuation coefficient Phase coefficient Wavelength Line length FLF of a line under length L at frequency f

Subscript denoting parameters at node S or node R Subscript denoting the type of disturbance source and response Subscript denoting parameters at f0 Subscript denoting maximum values of parameters Subscript denoting the standard limit of parameters Absolute values of variables

S, R p,q

0 M lim

|•| Acronyms UDTLs HWLLs FLF HTCs SOL

Ultra-long-distance transmission lines Half-wavelength transmission lines The Frequency-length factor Harmonic transmission characteristics Safe Operation Length

resonance frequencies of an UDTL in a fast manner, where various operation modes of the system are able to be considered. The rest of this paper is organized as follows: Section 2 induces the analysis model that takes FLF as a variable for HTCs analysis of UDTLs. Section 3 studies HTCs of standard HWTLs based on the pole distribution of FLF. In Section 4, the HTCs of other UDTLs including nonstandard HWTLs are analyzed. The effects of installing reactive power compensation devices are discussed in Section 5. Several cases are studied in Section 6 to verify the effectiveness of the proposed method. Finally, conclusions are drawn in Section 7.

(c). Refs. [14–16] investigate the transmission characteristics of long transmission lines based on the distributed-parameter line model. However, they all assume open-ended lines and do not discuss the impact of load condition. (d). Ref. [17] explores the propagation rule of wideband frequency harmonic in power grid. Ref. [18] uses a numerical approach to calculate the harmonic impedance of UDTLs in the resonance analysis. Nonetheless, the relationship between parameters and HTCs are not clear. (e). Ref. [19] uses Resonance Mode Analysis method (RMA) to identify the harmonic resonance and impacts of various network components on the resonance for transmission grids with cables penetrated. In addition, Ref. [20] performs simulations to find the 3rd and 5th harmonics characteristics of UTDLs. Nevertheless, a significant large volume of calculation and simulation should be done to obtain the HTCs when using these methods. None of the above methods can describe the HTCs of UDTLs clearly, and take all parameters, such as line loss, operation modes and line length, into consideration. Therefore, a specific analysis method is needed to be established. To investigate the HTCs of UDTLs, this paper proposes a method that considers the comprehensive effect of various parameters, i.e. frequency, line length, line loss, and the operation mode of the system. First, the Frequency-Length Factor (FLF) of UDTLs is innovatively proposed as an indicator to reflect the simultaneous effect of frequency and line length. Then, the frequency transmission models of lossy UDTLs are derived as functions of FLF. The relationships between each parameter and variables in functions are clear and concise, which makes it possible to reveal HTCs. The HTCs of HWTLs are first evaluated based on the pole distribution characteristic of FLF. Next, the severity of harmonic amplification is calculated by evaluating the safe operation length, as a solution to mitigate power quality issues. Finally, an approach simplifies the repeated spectrum analyses into the calculation of intermediate variables to estimate all potential harmonic

2. Models of frequency transmission 2.1. Hyperbolic function model of transmission line The simplified topology and the frequency model of an UHVAC system are shown in Fig. 1. The disturbance sources in power grid, including current and voltage harmonic sources, can be analyzed separately according to superposition principle [21]. The distributed parameter model is applied on UDTLs, as shown in Fig. 1(b). Based on the two-port network of transmission line, UR and IR can be expressed as

⎧ UR = US ch (γf L) − IS Zc sh (γf L) I ⎨ IR = IS ch (γf L) − S sh (γf L) Zc ⎩

(1)

The hyperbolic functions can be expanded to plural form as

⎧ ch (γf L) = ch (αf L) cos (βf L) + jsh (αf L) sin (βf L) ⎨ sh (γf L) = sh (αf L) cos (βf L) + jch (αf L) sin (βf L) ⎩

(2)

As seen from Eq. (2), the HTCs of UDTLs are related to the product of βf and L. Therefore, βf L is defined as Frequency-Length Factor (FLF), which is expressed as 2

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Table 1 Parameter expressions of calculation models. p

q

Mpq

Apq

I

i

|n|

−[0.25(1 − |n|2 )2 + b2]0.5

U

u

1

+[0.25(1 − |n|2 )2 + b2]0.5

U

i

|n|/|Zc |

+[0.25(1 − |n|2 )2 + b2]0.5

I

u

|Zc |

−[0.25(1 − |n|2 )2 + b2]0.5

|Tpq | =

Mpq Apq sin(2νf , L + φ) + B

(8)

where Mpq and Apq are shown in Table 1. 2.2.2. Pole distribution characteristics of FLF The harmonic transmission models are simplified to trigonometric functions of FLF by Eqs. (1)–(8). It is worth mentioning that, although B is relevant to FLF as well, it changes little in the case of f < 1250 Hz and L < 3000 km. Therefore, the effect of B on the pole distribution characteristics of FLF can be neglected. On the basis of Eq. (8), the extreme value conditions of transfer functions for current and voltage disturbance injection, i.e. the pole distribution characteristics of FLF, can be calculated by

Fig. 1. Simplified UHVAC power system: (a) topology. (b) Frequency model.

f L νf , L = 2π⋅ ⋅ f0 λ 0

2νf , L + φ =

(3)

As seen in Eq. (3), the FLF is a variable in radians and related to the product of f and L. It reflects the phase change in HTCs.

2νf , L

(4)

UR = IR × ZR

From Eqs. (1) and (4), the current transfer function can be expressed by associating the response current at node R and the injected current disturbance source at node S, as follows

I n TIi = R = IS nch(γf L) − sh(γf L)

3.1. Characteristics of harmonic current transmission 3.1.1. Current response According to Eq. (3), the FLF of a standard HWTL (νh, λ 0 /2 ) is hπ at harmonic frequencies. Otherwise, FLF is fractional multiple of π at inter-harmonic frequencies. Accordingly, the transmission coefficient of harmonic current can be expended based on Eq. (8) as

(5)

For convenience, let n = a − jb , where a is larger than 0. The value a is directly proportional to active power (P). Similarly, b is positive in an inductive power system and directly proportional to Ssc. Expanding Eq. (5) in plural form by Eq. (2), it obtains

|TIi | = |n|/ AIi sin(φ) +

where αf reflects the effect of harmonic line loss. To simplify Eq. (11), this paper considers standard HWTLs neglecting line loss (αf = 0 , L = λ 0 /2 ). From Eq. (1), it can be obtained that |TIi |αf = 0 = 1. Substituting this equation into (11) yields

[(ach(αf L) − sh(α f L))cosνf , L + bsh(αf L)sinνf , L)] +j[(ach(αf L) + sh(αf L))sinνf , L − bsh(αf L)cosνf , L ]}

(6)

By successive simplification, the current transmission coefficient at f can be expressed as

|TIi |αf = 0 = |n|2 / AIi sin(φ) +

MIi AIi sin(2νf , L + φ) + B

a2 + b2 , AIi =

+ b2 , φ = arctan

(

1 − |n|2 2b

)

1 + |n|2 ≡1 2

(12)

thus (11) can be simplified by (12) as

(7) (1 − |n|2 )2 4

1 + |n|2 + (1 + |n|2 )sh2 (αf L) + ash(2αf L) 2 (11)

n

where MIi = |n| =

(10)

To explore HTCs of UDTLs, standard HWTLs of which lengths are half a wavelength of the fundamental power frequency are discussed in this section.

At node R in Fig. 1(b), there is

|TIi | =

(9)

π + φ = − + 2N π 2

3. Harmonic characteristics of standard HWTLs

2.2. Model of current disturbance transmission

TIi =

π + 2N π 2

|TIi | = 1/ 1 +

,

1 + |n|2

B = ⋅ch(2αf L) + ash(2αf L) . The φ should satisfy −π/ 2 2 < φ < π/2 in a power system without short circuit or no load. Otherwise, φ should be either −π/2 or π/2 when short circuit or no load occurs on the receiving system, respectively.

1 + |n|2 2 a ⋅sh (αf L) + 2 sh(2αf L) |n|2 |n|

(13)

Theoretically, it can be concluded from Eq. (13) that 1) when αf = 0 , |TIi | equals 1 at harmonic frequencies. 2) when αf ≠ 0 , |TIi | at harmonic frequencies is related to factors such as αf , n, etc. The increase of P or Ssc reduces |TIi |. Additionally, since both |n|2 and a are greater than 0, |TIi | of standard HWTLs is always smaller than 1.

2.2.1. Frequency transmission models of AC UDTLs Using a method similar to the one outlined previously, all equivalent frequency models for AC UDTLs can be simplified to 3

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3.1.2. Voltage response The electric power utilities supervise harmonic issues mainly by observing voltage components. However, there is no clear relationship between the amplification of the response current and that of the response voltage. As a result, further considerations should be given to the characteristics of harmonic voltage response, which is the real concern of the power system analysis. A method based on the reference impedance is therefore proposed to investigate whether the response voltage is amplified. The core idea is as follows: The impedance seen at the sending end of a lossy standard HWTL at f0 is written as ZS0 = RS 0 + jXS 0 . Then, the harmonic impedance of ZS0 neglecting the coupling between the transmission line and the grid is defined as the reference impedance (Zre = h RS0 + jhXS0 ). If |ZS | > |Zre |, the harmonic response voltage at sending end (US = IS |ZS |) is greater than the reference voltage (Ure = IS |Zre |), which means US is amplified. Then, US is transferred to the receiving end, resulting in a response voltage UR . This response voltage can be attenuated compared to US . However, as long as it is greater than Ure , the response voltage is amplified. Accordingly, the amplification criterion of the response voltage produced by current disturbance injection can be expressed as * TIu

Fig. 2. Relationship between fr and SSC.

νf , L = −

U = R >1 Ure

(14)

1 + nth(αf L) n + th(αf L)

(15)

Therefore, the impedance seen from node S and R to the load at f0 1 + n th(α L) Z can be expressed as ZS0 = Zc n +0th(α 0L) and ZR0 = n c ≈ ZS0 , respec0

0

0

tively. Since α 0 is small, it can be simplified that Zre ≈

ZR ≈

ZL2 ⋅ X sys 2 hZL2 + hXsys

. As a result, K =

|ZR | |Zre |

≈

1 + K c2 , h + K c2

ZL2 ⋅ X sys 2 hZL2 + Xsys

(K + k ) f

1 Z P arctan ⎡ 2(K + k ) f f − c 2f 2 0 − 2(K +c k ) V 2 ⎤, where k2 = 6.7 Q −0.74, P c 2 0⎦ ⎣ c 2 0 and the CIGRE load model in Ref. [22] is chosen. Accordingly, φ inφ φ π π creases with f, so ΦI = 4 − 2 and ΦU = − 4 − 2 decreases with f. Then, as shown in Fig. 2, each curve of ΦI and ΦU goes up (from ΦI to Φ2 ) with the increase of SSC. Therefore, their intersection points with the line of νf , L move to the right (from fr1 to fr2), which means fr. increases.

and

which is smaller than 1.

Moreover, K is a constant if K c is fixed. * is smaller than 1 at harmonic frequencies, which Accordingly, TIu means that the harmonic response voltages produced by harmonic current injections cannot be amplified on standard HWTLs.

3.2.2. Resonance magnitude and its sensitivity The envelope of local maximum values of transmission coefficient for disturbance current and voltage can be expressed as Eqs. (19) and (20), respectively. It can be demonstrated that b increases with Ssc , resulting in a larger TIM when K c is fixed. On the contrary, TUM decreases with Ssc .

3.2. Characteristics of harmonic voltage transmission Similar to the approach in Subsection 3.1, it can be obtained that the |TUu | of a lossless standard HWTL at harmonic frequencies is 1. Then, there is AUu sin(φ) + fied to

1+ | n|2 2

≡ 1 as well. Therefore, |TUu | can be simpli-

|TUu | = 1/ 1 + (1 + |n|2 )⋅sh2 (αf L) + ash(2αf L)

(16)

Accordingly, similar conclusions as those in Subsection 3.1 can be drawn, as follows: if the line loss is neglected, |TUu | equals 1 at harmonic frequencies. Otherwise, |TUu | is smaller than 1. That is, there is no amplification in harmonic voltage propagation for standard HWTLs in any cases considering line loss. Moreover, the increase of P or Ssc leads to larger |TUu |.

φ π − + Nπ 4 2

1 a * TIM = |TIu |max = K/ ⎛1 + 2 ⎞⋅sh2 (α f L) + 2 sh(2α f L) |n| ⎠ |n| ⎝

(19)

TUM = |TUu |max = 1/ (1 + |n|2 )⋅sh2 (αf L) + ash(2αf L)

(20)

4. Transmission characteristics of other UDTLs 4.1. Models of harmonic transmission In practice, the lengths of the transmission lines deviate from that of standard HWTLs (λ 0 /2 ) due to economic factors. Considering the span of power transmission, UDTLs with lengths that differ λ 0/2 by less than ± 20% are especially defined as non-standard HWTLs [23,24]. It is proved that harmonics cannot be amplified on standard HWTLs. However, this does not mean that non-standard HWTLs have the same feature, although the length of them are very close to λ 0 /2 . It is derived for standard HWTLs that, νh, λ 0 /2 = hπ at harmonic frequencies. Based on this feature, the FLF of other UDTLs at harmonics

3.2.1. Resonance frequency and its sensitivity According to the pole distribution characteristics of FLF, the transmission coefficient reach local maximum values when νf , L satisfies one of the conditions as follows

νf , L =

(18)

Eqs. (17) and (18) are corresponding to the case of current and voltage disturbance injection, respectively. According to the definition of φ , the right side of Eqs. (17) and (18) are both non-integral multiple of π in normal operated systems. However, in accordance with Eq. (3), only when disturbance source is interharmonic can νf , L of standard HWTLs be fractional multiple of π. Therefore, it can be concluded that only inter-harmonic disturbances can cause resonance on standard HWTLs. Based on Eqs. (17) and (18), it can be obtained that fr increases with φ The reason is that, by simplification, ≈ Ssc .

|Z |

where UR can be written as |TIi |⋅ |ZR | . It is verified that |TIi | < 1 at harUre re monic frequencies. Therefore, the next step is to determine the relationship between |ZR | and |Zre |. Based on Eq. (1), the harmonic impedance of a transmission line with length L seen from node S to the load can be simplified as

ZS = Zc

φ π − + Nπ 4 2

(17) 4

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vary with L* as a trigonometric function for the studied harmonic. Therefore, the beams of the extreme values are formed with the change of Ssc . As a result, an envelope curve ( TIE ) that reflects the local maximum response voltage can be obtained. In UHVAC power systems, the 3rd, 5th, 7th, 11th and 13th harmonics are the dominant harmonic sources, of which the individual voltage content can generally reach about USM = 0.5%. This voltage component can be regarded as a response produced by injecting current disturbances at the sending end with a percentage of ISM=USM/Zre. This * ⋅USM . makes the response voltage at the receiving end UR =|TIu |⋅ISM=TIu Although harmonics can be amplified in UDTLs’ propagation, the amplification is acceptable if UR is smaller than the standard limit (Ulim ). The satisfactory length in this condition is defined as the Safe Operation Length (SOL) for the analyzed harmonic. Otherwise, the response voltage exceeds the standard limit when UR > Ulim . The criterion of current U * is the amplification can also be written as TIu >Tlim , where Tlim= I ⋅lim SM |Z re | limit of transmission coefficient. The SOL region of each harmonic current can be calculated by the aforementioned approach. The length of a transmission line should be within this region to avoid over-limit amplification. Otherwise, there is potential risk of severe harmonic amplification for power grid. By using a similar method, the SOL region can be calculated for harmonic voltage source injection, of which the transmission coefficient limit is Tlim=Ulim/USM . If TUu >Tlim , the injected harmonic voltage can cause over-limit amplification. There are several harmonic limitation standards [25–27]. Among these, IEEE STD.519 is more conservative than the others for UHV systems, thus is taken as the reference standard.

can be written as Eq. (21) to describe harmonic characteristics of nonstandard HWTLs intuitively.

λ νh, L = βh ⎡ (1 + L*) 0 ⎤ = hπ + Δνh, L* 2⎦ ⎣ where Δνf , L

(

= L− λ0 . 2

λ0 2

* * = h πL

)( ) /

λ0 2

(21)

is defined as the derivative of FLF, and L*

is the relative length of the studied UDTL differs from

Therefore, the harmonic transmission models can be expressed as

|Tpq | =

Mpq Apq sin(2Δνh, L * + φ) + B

(22)

where Mpq and Apq can be referred to Table1. Additionally, ZS can be considered as linear neglecting the coupling of UDTLs. Therefore, the 2L reference impedance can be expressed as |Z re |= λ |Zre |λ 0/2 , where |Z re |λ 0/2 0 is the reference impedance of standard HWTLs. Noticing that Δν h,L* is defined at harmonics, Eq. (22) is used as an additional method of HTC analysis that only considers harmonics. Otherwise, if inter-harmonics are involved, Eq. (8) is also a necessity in analysis. For instance, Eq. (8) can be applied to calculate resonance frequencies, whereas Eq. (22) is only able to figure out the frequencies of the local maximum values of transmission coefficient among all harmonics. 4.2. Harmonic Transmission Characteristics Eqs. (8) and (22) are applicable for both non-standard HWTLs and all other UDTLs. Meanwhile, since both functions have the same structure, similar conclusions can also be drawn from Eq. (22) as that from Eq.(8) in Section 3. Accordingly, the HTCs can be obtained by jointly exploiting both of them, as follows 1) Estimation of harmonic resonance frequency The workload of traditional methods is huge when calculating all potential harmonic resonance frequencies under various operation modes for a power system with UDTLs. However, based on the pole distribution of FLF, the repeated spectrum analyses are simplified into the calculation of intermediate variables (ϕ and ν h,L ). The estimation procedure is shown in Fig. 3. 2) Amplification: The harmonic transfer coefficients of UDTLs are trigonometric functions of Δνf,L* . Therefore, |Tpq | can be larger than 1 instead of always being smaller than 1. That is, harmonic current or voltage can be amplified on all UDTLs except standard HWTLs. Even non-standard HWTLs, of which the length is very close to λ 0 , are not exempted. 2 Eqs (13) and (16) are tenable when νf,L=Nπ . As a result, current or voltage disturbances cannot be amplified on such lines. For example, even harmonics cannot be amplified when propagating on a L=λ 0/4 transmission line. 3) Local maximum values: The local maximum values of harmonic transmission coefficient of UDTLs are the same as Eqs. (19) and (20). It can be therefore concluded that a larger P or Ssc leads to a bigger TIM and a smaller TUM . 4) Pole distribution: The extreme condition of Eq. (22) is Δνf,L*=ΦI + Nπ and Δνf,L*=ΦU + Nπ for current and voltage disturbance injection, respectively. In addition, the frequencies that lead Eq. (22) to reach local maximum values increase with P and Ssc .

5. Discussion on the effect of installing reactive power compensation devices HWTLs present specific properties for bulk power transmission over long distances. They have good transmission stability performance [28]. That is, the voltage in the middle of a HWTL is proportional to the ratio of the transmitted power and its surge impedance load, while the voltage magnitude is almost equal at both sending and receiving ends [29,30]. The reactive power of the HWTL can also be automatically balanced. Due to these properties, there is no need for additional reactive power support [31], making the calculation model

1 1 ⎞ n = Zc ⎛⎜ + ⎟ Z Z + Z sys L T ⎠ ⎝

(23)

However, Ref. [32] mentions that reactive power optimization and voltage control is needed for HWTL systems to reduce partial overvoltage caused by overload or low power factor. Additionally, to reduce resistive power losses and control system voltage levels for other UDTLs, reactive power compensation devices are deployed in power systems [33]. Here, a typical reactive power compensation strategy, i.e. the shunt capacitors connected at the receiving end, is considered.

4.3. Severity of harmonic amplification From 4) and 3), the effects of P and Ssc on |TIu | and |TUu | are obtained. These can be used to determine the severity of harmonic amplification. Here, the severity of harmonic amplification indicates whether the amplification is acceptable. As an example, Fig. 4 depicts the 3-D concept diagram in the case of harmonic current injection. As shown in the projected platform, |TIu |

Fig. 3. Estimation procedure of harmonic resonance frequencies. 5

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b, leading to the decrease of fr. Many other reactive power compensation strategies have been proposed [34,35]. Their advantages with regard to maintain good power quality are accompanied by complicated electric circuit structures. This can cause difficulty in HTCs analyses, which needs to be further discovered. 6. Case study To verify the accuracy of the proposed method, the single-line diagram of an UHVAC system with an UDTL connected is built in PSCAD as a reference. The characteristics of standard HWTLs, nonstandard HWTLs and three UDTLs are studied. Comparisons between the calculation of the proposed harmonic transmission model and the PSCAD simulation are made. 6.1. Transmission characteristics of standard HWTLs * The spectrum of TIu and |TUu | of a standard HWTL within 750 Hz is illustrated in Fig. 5, where the SSC of Mode A (MA) is 50 GVA, and Pn = 4066 MW. Table 2 shows the typical parameters of UHVAC transmission lines [36]. As observed from Fig. 5, fr increase with SSC in both cases. When SSC goes up, the resonance transmission coefficients of current increase, but those of the voltage decrease. Table 3 shows the transmission coefficients at harmonic frequencies * | and TUu ) and at resonance frequencies (TIM and TUM) within 1250 (|TIu Hz. Calculation errors of these coefficients and fr compared to the simulation results are also listed. Conclusions can be drawn from Table 3 that harmonics cannot be amplified neither can it cause resonance, whereas inter-harmonics can be amplified by about 10 times. In addition, the error of calculation is less than 0.1%, which means the proposed models have satisfactory accuracy.

Fig. 4. Illustration of transmission coefficient with SSC and L*.

When shunt capacitor units are connected at the receiving bus, n in the calculation model should be written as

1 1 n=Zc ⎜⎛ + + jXCP ⎞⎟ Z Z ZT + L ⎝ sys ⎠

(24)

Compared with Eq. (11), there is a new variable jZc XCP in n. Therefore, there is

b=

Zc Ssc − hXCP Zc hV 2

(25)

It can be seen that shunt capacitors could couple with receiving system, making HTCs of UDTLs more complicated. However, some of the features can still be obtained based on Eq. (24), as follows 1) The value of b decreases with f, and equals 0 when f =

6.2. Harmonic amplification severity of non-standard HWTLs

Ssc f. XCP V 2 0

To obtain the impact of operation mode on SOL, the SOL region of four operation modes that vary from no-load to full-load are analyzed. These operation modes including Mode A, Mode B (MB), Mode C (MC) and Mode D (MD). MB represents the system with Ssc = 100 GV and P = 4000 MW. MC represents the system with Ssc = 50 GV and P = 2000 MW. MD is the system with Ssc = 100 GV and P = 2000 MW. The result of the 3rd harmonic is shown in Table 4, where the SOL regions are given as intervals. It can be obtained that the larger the Ssc

2) Since φ decreases with b, it increases with f. Therefore, ФI and ФU still decrease with f. As a result, the sensitivity of resonance frequency is the same as that of HWTLs in Subsection 3.2.1. That is, fr increases with SSC. 3) Compared with systems without any reactive power compensation devices, the connection of shunt-capacitor units deducts the value of

Fig. 5. Transmission coefficient with frequency: (a) current disturbance source. (b) Voltage disturbance source. 6

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Table 2 Typical parameters of UHVAC transmission lines.

Table 5 Harmonic amplification severity.

R0 (Ω/km)

L0 (mH/km)

C0 (μF/km)

Pn (MW)

λ0 (km)

0.00783

0.8330

0.01379

4066

5898

h

MA

1 4

Current

* |TIu | TIM

0.88–0.95

0.64–0.98

0.37–0.99

2.2–13.8 < 1%

0.9–13.34 < 1%

0.42–9.7 < 1%

< 0.2%

< 0.2%

< 0.2%

0.92–0.98 2.19–3.63 < 0.1% < 0.2%

0.98–0.99 6.88–5.7 < 0.1% < 0.2%

0.99 6.83–10.2 < 0.1% < 0.2%

Voltage

Errors Errors TUu TUM Errors Errors

* | and TIM of |TIu of fr

of TUu and TUM of fr

MA

1 10

MA

TUM

13

−11.36 0.08%

−8.48 0%

−14.92 −0.06%

−12.79 0.15%

(%)

Calculation Error Calculation Error

12.54 −0.09% −15.86 0%

8.95 0.05% −9.7 0%

6.99 0% −6.99 0%

5.159 0.04% −13.57 0.07%

4.396 0.1% −19.18 0.1%

Calculation Error

13.08 0.02%

10.71 0.03%

9.326 −0.04%

8.478 −0.08%

8.477 −0.08%

Table 6 Harmonic resonance orders and transmission coefficient. L h

TIM

Table 4 SOL region of 3rd harmonic current and voltage.

h

TUM Type

11

−18.04 1.6%

* LM

Operation mode

7

Calculation Error

TIM

Source type

5

(%)

* LM

Table 3 The comparison between calculation and simulation results.

3

Source

Calculation Simulation Calculation Error Calculation Simulation Calculation Error

500 km

650 km

1500 km

2,6,7,8,12,13 2,6,7,8,12,13 4.3–34.2 0.2%–0.6% 4,5,9,10,11, 4,5,9,10,11, 8.54–45.93 0.01%–0.08%

2,5,6,9,10,11 2,5,6,9,10,11 1.3–53.06 0.3%–0.9% 3,7,8,12 3,7,8,12 12.65–34.21 0.02%–0.1%

2,4,6,8,10,12 2,4,6,8,10,12 2.18–4.832 0.2%–0.5% 3,5,7,11,13 3,5,7,11,13 10.55–26.73 0.05%–0.12%

SOL (%)

MA MB MC MD

Current

Voltage

(−15.85,1.32) (−14.47,1.32) (−15.85,1.32) (−14.47,1.32)

(−3.19,11.27) (−2.8,11.27) (−3.14,11.27) (−2.83,11.27)

or P, the smaller the SOL region. Moreover, when Ssc increases from 50GVA to 100GVA, the SOL region decreases by 2%–10%. However, compared to Ssc, the change of P has little effect on SOL region. Therefore, Ssc can be considered as the dominant factor that effects SOL. In addition, since SOL region decreases with SSC, the result of Ssc = 100 GVA is relatively conservative to ensure accuracy and to keep a certain margin. Meanwhile, the short circuit capacity of a UHV system in practical can reach about 50GVA. Therefore, the result of Ssc = 100

Fig. 6. SOL region of each harmonic: (a) current harmonic injection. (b) Voltage harmonic injection. 7

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C. Chen, et al.

sources to ideal voltage and current source models. However, with the increasing integration of renewable energy, more and more power electronic devices are connecting to power grid. The complex frequency characteristics of wind or solar power generations, and the coupling between these generations and power grid, can complicate the resonance mechanism. In the future, the HTCs of UDTLs will be further analyzed with the integration of renewable energies. The control strategies of wind turbines and solar photovoltaic systems will be considered in the HTC analysis.

GVA can be regarded as a practical guidance for the enhancement of power grid safe operation, as shown in Fig. 6. Table 5 lists the maximum transmission coefficients of each harmonic source injection and the corresponding length under MB. The results of calculation are compared with the PSCAD simulation output. It is seen that the over-limit amplification on non-standard HWTLs caused by harmonic current injection exceeds the standard limit by 1.2–3.5 times, and that from harmonic voltage injection exceeds by 2.2–4.5 times. Moreover, the calculation error is less than 1%. The accuracy of the proposed model is verified. To avoid the harmonic over-limit amplification on non-standard HWTLs, the selection of line length can be quantitatively adjusted by Fig. 6. For instance, if the distance between the sending and receiving power system is about 3055 km (L* = 4%), and the 3rd harmonic current is the dominant disturbance source, the length of this HWTL should be shortened slightly when construction. Changing the length to 3011 km (L* = 2.5%) can ensure the 3rd harmonic response voltage be within the limit. In addition, if the length of a non-standard HWTL is in the range of 2821 km < L < 2985 km (−3.95% < L* < 1.6%), there is no current over-limit amplification. The above length region is called “absolute SOL region” of harmonic current injection. Similarly, 2893 km < L < 3016 km (−1.51% < L* < 2.66%) is the “absolute SOL region” of harmonic voltage injection. Lines within absolute SOL region can be operated safely, and no over-limit amplification would occur.

Conflict of interest The authors declare that they have no conflict of interest. Authors contribution Chang Chen and Honggeng Yang conceived of the presented idea, and developed the theory. Weikang Wang and Wenxuan Yao performed the computations and verified the analytical methods. Mirka Mandich conducted the error analysis. Yilu Liu supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. References

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The lengths of UDTLs are selected according to reference [37–39]. The harmonic orders (h) and the range of transmission coefficient in cases of resonance are calculated. Table 6 lists the results that h < 13 for each UDTL. It shows that the proposed method can estimate all potential harmonics and transmission coefficients of resonance with error below 1%. In addition, it can be seen that several harmonics is amplified by up to dozens of times on a UDTL. 7. Conclusions and recommendations In this paper, the equivalent frequency transmission models considering line loss are established for harmonic amplification investigation in AC UDTLs. The estimated results will help power utilities develop strategies to mitigate harmonics and enhance power delivery safety. The main contributions of the paper can be summarized as follows 1) A method based on the pole distribution characteristics of FLF is proposed for harmonic transmission analysis of lossy AC UDTLs. This method considers the comprehensive effects of frequency, line length, and operation modes of the power system, revealing the mechanism of the harmonic transmission in AC UDTLs. 2) The proposed method mathematically proves that, when propagating on lossy standard HWTLs, only inter-harmonics can cause resonance. 3) The SOL of each harmonic is quantitatively analyzed. To avoid overlimit amplification and ensure the safe operation of power grid, the length within the SOL area should be analyzed first. 4) All potential frequencies of harmonic resonance for an AC UDTL can be fast estimated using the proposed method. Since the estimation considers different operation modes, it can be used as a guideline for transmission line design and the safety evaluation of existing lines. 5) By comparing the results of the derived models with PSCAD simulation, the accuracy of the proposed method on estimating transmission coefficients and resonance frequencies for lossy AC UDTLs is verified. The proposed method has great applicability for various AC UDTLs and system operation modes. A limitation of the proposed method is that it simplifies disturbance 8

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