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Analysis of electrical length compensation types for tuned half-wavelength transmission lines
Electrical Power and Energy Systems 115 (2020) 105520
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Analysis of electrical length compensation types for tuned half-wavelength transmission lines☆
T
Pulin Caoa, Hongchun Shua, Bo Yanga, , Shengnan Lib, Tingyi Heb, Lei Yangb, Yiming Hana, Tao Yuc ⁎
a
Faculty of Electric Power Engineering, Kunming University of Science and Technology, 650500 Kunming, China Electric Power Research Institute of Yunnan Power Grid Co., Ltd., 650217 Kunming, China c College of Electric Power, South China University of Technology, 510640 Guangzhou, China b
ARTICLE INFO
ABSTRACT
Keywords: Compensation circuit Frequency response Half-wavelength transmission lines (HWTL) Modal-phase transformation Power transmission
The compensation circuits are the core to improve the flexibility of practical application of half-wavelength transmission lines (HWTL), which are restricted to the fixed transmission length due to the constant fundamental frequency. This paper proposes a mode-phase transformation based method to design the parameters of compensation circuits after comparison of frequency responses of three kinds of compensation circuits, e.g., T segments, Π segments and L segments. At first, the compensation circuit parameters are obtained in the modal system that separates the coupling multiconductor system into isolated circuits to avoid the complexity of the three-phase system. Then, the compensation circuit parameters can be transformed into multiconductor system by modal-phase transformation, which accomplishes the compensation circuit parameter design in three-phase system. Lastly, the parameters acquired by the proposed method are compared with the natural HWTL. Comprehensive simulations are undertaken which validate that the proposed method is more suitable for tuning HWTL.
1. Introduction DUE to the carbon dioxide emission restriction of the United Nations Framework Convention on Climate Change [1,2], the worldwide demand of renewable energy, which is an ideal replacement of traditional fossil fuel, is apparently ever-increasing. In general, the renewable sources around load centers are insufficient to sustain the fast development of load centers in countries with continental extensions, such as Brazil, Russia, and China. In fact, the small-scale utilization of the renewable energy, which requires just some simple equipment, can be easily implemented near the low-capacity load such as the microgrid without strict demand of operating conditions [3]. However, the largescale renewable energy base requires strict climatic and geographic conditions, i.e., appropriate wind speed, annually stable sunlight and consistently abundant wind resource [4]. This dilemma always occurs at the countries with continental extensions, in which regional economic growth is notably unbalanced. For example, the Gangsu wind farm is the world’s largest wind farm with a capacity of 6800 MW and is expected to reach a total capacity of 20,000 MW by 2020, is located in
Gangsu province in the undeveloped area of northwest China with very few population and power demand. Nevertheless, Gangsu province is unable to consume all the capacity of this large-scale wind farm, hence most of its wind power has to be transmitted to the load centers located at east coast of China, e.g., Shanghai and Guangzhou, which is about 2600 km away from Gangsu. Therefore, the bulk power transmission over thousands of kilometers is undoubtedly inevitable to overcome the extremely long distance between large-scale power plants and load centers. To remedy the above challenge, ultra-high voltage direct current (UHVDC) systems and ultra-high voltage alternative current (UHVAC) systems have been regarded as a prominent solution for the bulk electric power transmission with extremely long distance in China and Brazil. In general, considerable reactive power compensation and protection scheme improvement is required for UHVAC when the length of transmission line is much smaller than half-wavelength, which has resulted in few applications of UHVAC [5]. Unfortunately, there is no real half-wavelength transmission lines in practice. Although the low capacity half-wavelength transmission lines
This work was supported in part by the National Natural Science Foundations of China under Grant 51807085, 51977102 and 61963020, and in part by the Yunnan Provincial Talents Training Program under Grant KKSY201604015. ⁎ Corresponding author. E-mail address: [email protected] (B. Yang). ☆
https://doi.org/10.1016/j.ijepes.2019.105520 Received 24 January 2019; Received in revised form 21 August 2019; Accepted 27 August 2019 Available online 02 September 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
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(HWTL) are widely applied in the communication engineering domain and the basic concepts of half-wavelength power transmission have been proposed since 1939 [6], the large capacity HWTL with ultra-high voltage are prudently considered in Russia, Brazil and China, which has not been implemented in practice due to the temporarily insufficient power demand and other technical problems such as its constant line length. However, such technique is prominent and worth investigating in future as the power demand is increasing rapidly while more advanced techniques appear. In the past decade, HWTL has gained significant interests around the globe thanks to the technological advancements, especially the improvements of ultra-high voltage transmission line operation and control. Compared to the widely applied high voltage direct current (HVDC) which is a mature technology nowadays for bulk power transmission over extensive geographical region [7], HWTL does not need expensive high-voltage converters, such that the installation costs could be reduced. Ref. [8] concludes that the investment for the single-circuit HWL with reverse phase can be considered 30% (1.15 billion USD) lower than that for HVDC and the total yearly cost is 15% lower on the condition of the power transmission of 6000 MW over 2500 km. In addition, the traditional singleend relay protections are unable to distinguish the remote-end faults from faults on other lines connected to the same bus without auxiliary devices due to the extremely long distance of HWTL. The nonlinear variation of the impedance induces that the short-circuit impedance of HWTL varies between infinity and zero, hence the traditional distance protection that needs to calculate the short-circuit impedance may lose efficiency in HWTL, thus the conventional protection is unable to applied directly for HWTL. Ref. [9] proposed a scheme with five kinds of protection to promptly identify the fault and cover the whole line from single-end in order to promptly identify the fault and cover the whole line by traditional protection. In addition, travelling wave based differential protection scheme and have been applied by literatures [10] for HWTL, which attempt to overcome the adverse effect of extremely long distance that may invalidate the relay protections. In order to promote the success rate of auto-reclosing of HWTL, literature [11] implemented a reactor to ground the power transformers after the fault occurring. Moreover, the long distance of HWTL is also quite inconvenient for the fault detection and search, thus a fault location technique based on travelling wave has been proposed for faults that generates obvious travelling waves [12]. However, the attenuation of high frequency components caused by faults or breaker operations are so severe in the long distance transmission, such that the high frequency components are usually difficult to be detected at bus station. Thus, Ref. [13] proposed a fault location method relying on symmetrical components. In order to achieve the characteristics of half-wavelength, the length of HWTL has to be fixed around 3000 km or 2500 km due to the constant fundamental frequency of 50 Hz or 60 Hz, [14], which therefore restricts the practical application of HWTL. In order to control the power flow of the HWTL, the series and shunt flexible ac transmission systems (FACTS) are applied in HWTL to control the main power flow [15], which demonstrates the feasibility of the FACTS to variate the power flow in HWTL. Based on the communication engineering, work [16] proposed a compensation algorithm to shorten a transmission line than half-wavelength to the tuned HWTL with compensation circuits consisted of inductance and capacitance. Additionally, three kinds of compensation circuit, e.g., T segment, Π segment and distributed shunt capacitors, have been developed, which concluded that the distributed shunt capacitors are less efficient than T segment and Π segment formed compensation circuits. Compared with HVDC, literature [17] analyzed the flexible application of the tuned HWTL, which shows that the transmission efficiency of the tuned HWTL is similar to that of HVDC. Moreover, work [18] attempted to increase the flexibility of HWTL by generating high fundamental frequency from 180 Hz to 360 Hz to make HWTL suitable for the short-distance transmission, in which the length of HWTL can be reduced to less than 1000 km.
Although the frequency conversion significantly decreases the length of HWTL, an AC-AC or an AC-DC-AC conversion at the ends of transmission lines is required to transform high frequency power to 50 Hz or 60 Hz component for traditional fundamental frequency loads, which requires high voltage and large capacity electronic devices as HVDC. Thus, high costs of electronic devices are inevitable in this method, which significantly increases the overall investment. Although the compensation types for tuned half-wavelength transmission lines have been presented in the Ref. [16], the necessary of the capacitance between phases should be further considered. On the other side, the segments of T, Π and L have been depicted in other Ref. [19], but their advantages and the differences for compensation circuits, especially their frequency responses need to be analyzed. Furthermore, the diverse compensation types and their cascades replace the actual transmission lines, of which only the influence at the stable voltage distribution on the transmission lines is studied [20]. Due to the outstanding performance at stable voltage distribution on the transmission lines, a large quantity of cascades of the simple compensation circuit is believed to be an excellent choice to replace the transmission lines. Hence whether a large number of cascades of segments is necessary for the compensation circuit should be studied. The reduction of the actual length of transmission lines is the main advantage of the tuned HWTL. Being similar to the natural HWTL is able to restrict the technical problems similar to the natural HWTL. If the compensation circuit changes the unique characteristics of HWTL, more problems, such as voltage distribution along lines and phase variation, have to be considered, which further increase the complexity of the technical problems. Thus, the similarity of the compensation circuit and transmission lines is critically necessary. In this paper, the frequency responses of compensation circuits formed by T segment, Π segment and L segment are analyzed to present the frequency response variations resulted from the different segment numbers and structures. Moreover, the compensation circuits regarded as two-port circuits are firstly designed in the modal system. Then the compensation circuit parameters are transformed into three-phase system for compensation circuit design. Lastly, comprehensive case studies are carried out to verify the effectiveness of the proposed compensation-circuits in comparison to the traditional compensation circuits. The main contributions of this manuscript can be summarized as: (1) Considering the different parameters of air mode and earth mode, this work proposes a compensation parameter calculation scheme, which calculates the parameters of inductance and capacitance in mode system and then transforms the parameters to the phase mode to complete the parameters of the compensation circuit design. The mutual inductance and capacitance between phases are fully taken into account in this design scheme, which makes the designed compensation circuit more similar to the transmission lines than the other compensation circuits. (2) Due to the outstanding performance at stable voltage distribution at the transmission lines, a large quantity of cascades of the simple compensation circuit is believed to be an excellent choice to replace the transmission lines. The transient waveform of the tuned halfwavelength transmission lines is analyzed in this manuscript to verify that the number of the cascades of the compensation circuit should be restricted to avoid the oscillations induced by compensation circuits. (3) Based on the frequency response analysis, the structure of Π segment is verified to be a better compensation circuit unit than the other two that may amplify some frequency components. 2. Equalization of lossless two-port transmission lines In order to accurately represent the transmission lines by lumped circuits, the lossless model, Π and T models, are often adopted. In addition, they are represented by an infinite series of the L model in 2
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Table 1 Positive sequence parameters of 1000 kv transmission lines. Resistance R1 (Ω/km)
Inductance L1 (H/km)
Capacitance C1 (F/km)
Fundamental frequency f (Hz)
1.0324 × 10−2
8.4042 × 10−4
1.3752 × 10−8
50
unnecessary. Although the frequency response at fundamental frequency is the core of compensation circuit for power transmission, the frequency responses of other frequency components, especially the poles and zeros that may cause relay protection failure and overvoltage, should also be studied. The lossless compensation circuit for HWTL illustrated in Figs. 1 and 2 have the same structure of the low pass constant k filter which consists of a ladder network of identical sections of passive circuit components [21]. Hence, the theory of passive analog filter can be performed to calculate and analyze the frequency responses of the compensation circuit. The positive sequence parameters of the transmission lines which are implemented in 1000 kV ultra-high voltage Jingdongnan-NanyangJingmen lines in National Power Grid in China are tabulated in Table 1. The propagation constant γ of transmission lines can be calculated by [10]
Fig. 1. Compensation circuit models (a) Π model, (b) T model, (c) L model.
transient analysis. Thus, three kinds of lumped circuit models can be adopted for equalization of the transmission lines to increase the electrical length. In Fig. 1, l is the length of transmission lines; Lline and Cline are the line inductance and capacitance in per unit (p.u.), respectively. Then, each aforementioned equivalent model can form composite section by cascading n Π, T and L segments, which increase the similarity of the compensation circuit and distribution model of transmission lines, as demonstrated in Fig. 2. In the extreme case, the compensation circuit depicted in Fig. 2 can be regarded as transmission lines if the inductance and capacitance are infinitesimal. It composes an infinite series two-port circuit which are the fundamental hypothesis of lossless telegrapher equations. Since the segments number cannot be infinite in practice, the frequency responses of the compensation circuit are quite different from that of transmission lines. In order to properly equalize the normal power transmission and electromagnetic transients in the transmission lines, the frequency responses of compensation circuit should be much similar to that of transmission lines. However, it can only be obtained if the compensation circuit has the same frequency-dependent parameter to transmission lines, which is usually impossible due to the different structures and materials of transmission lines and lumped circuit. On the other hand, the compensation circuit is implemented to increase the electrical length at fundamental frequency to half-wavelength. It requires the magnitude attenuation at fundamental frequency to be as small as possible and the phase shifting to be the same to transmission lines. Hence, the accurate equalization of transmission lines spectrum is
=
+j =
(1)
(R1 + j L1)(G1 + j C1)
where α and β are the attenuation constant and phase constant, respectively; R1, L1, G1 and C1 are the resistance, inductance, conductance and capacitance per unit of the transmission lines. The attenuation constant and phase constant can be obtained by the equation above, as
=
1 [R1 G1 2
=
1 [ 2
+
(R12 +
2L 2 )(G 2 1 1
+
2C 2 ) ] 1
R1 G1 +
(R12 +
2L 2 )(G 2 1 1
+
2C 2 ) ] 1
2L
2L C 1 1
1 C1
(2) (3)
The conductance of the transmission line is small when compared to other parameters, and is neglected in these equations. The attenuation constant α, phase constant β and characteristic impedance Zc at fundamental frequency can be calculated by
=
1 [ 2
2L
=
1 [ 2
2L C 1 1
Zc =
1 C1
+
(R12 +
+
(R12 +
2L 2 ) 1
2L 2 ) 1
2C 2 ) ] 1
2C 2 ] 1
R1 + j L1 = 247.2 + j4.832 j C1
= 2.088 × 10
5
NP/km
= 1.068 × 10 - 3 rad/km
(4) (5) (6)
where ω = 2πf. The compensation circuit is aimed to increase the electrical length for transmission lines near half-wavelength, hence the extended electrical length is not supposed to be too large. Assume the electrical length Len that needs to be compensated is 100 km, the magnitude decay coefficient D and the time delay Td of the transmission lines at fundamental frequency are described by
D = e Len × 100% = 99.8%
Td =
2 f
Len = 340 µs
(7) (8)
Note that this paper selects 100 km compensation circuit as an example to increase the electrical length for transmission lines near halfwavelength, in which the extended electrical length is not supposed to be too large. The compensation circuits are comprised by 1, 2, 5 and 10 segments. Meanwhile, the characteristic impedance Zc is chosen as the
Fig. 2. Composite sections of compensation circuits (a) Π segments, (b) T segments, (c) L segments. 3
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Fig. 3. Magnitude responses of compensation circuits (a) Π segments, (b) T segments, (c) L segments; phase shifting of compensation circuits: (d) Π segments, (e) T segments, (f) L segments.
Fig. 3. (continued)
by increasing the segment number. However, it may cause severer signal distortion, which degrades the reliability of protection relay or fault detection equipment in the substations.
termination. The output magnitude and the delay of frequency components at termination are given in Fig. 3. It can be observed that the decay magnitude of fundamental frequency of each compensation circuit is almost zero. Meanwhile, the time delay at fundamental frequency of each compensation circuit is 340 μs, which are the same to the transmission lines. Therefore, all the compensation circuits are suitable to replace the transmission lines for power transmission at fundamental frequency. However, the magnitude and the number of the frequency components, which can be amplified by compensation circuits to induce overvoltage, increase in both L and T compensation circuits as the segments number grows. Therefore, the Π segments formed compensation circuit, which does not increase any frequency component magnitude is more suitable to equalize the transmission lines. Furthermore, the passband of compensation circuit can be extended
3. Compensation circuits for multiconductor transmission line 3.1. Phase-mode transformation for lossless multiconductor transmission lines Although the compensation circuit can be easily designed for the two-port transmission lines, the power grid for bulk AC power transmission is multiconductor transmission lines with three parallel conductors, which cannot be analyzed as simple two-port transmission lines. The relationship of primary line constants, voltage and current in lossless three-phase transmission lines can be written as follows 4
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iphase x
uphase x
= C phase
= L phase
uphase t
3.2. Parameter calculation in phase lines
iphase
(10)
t
The compensation circuit parameters are calculated to compensate the electrical length of different modal circuits. Then, all the parameters in modal circuits should be transformed to phase conductors to accomplish the compensation circuit design. The inductance Lm_com and capacitance Cm_com of the compensation circuit in the modal circuits can be transformed to phase system by
where
iphase = [ iA iB iC ]T
(11)
uphase = [uA uB uC ]T
(12)
C phase =
C0 + 2C C C C C0 + 2C C C C C0 + 2C
(13)
L M M L phase = M L M M M L
(14)
C mode = T 1C phase T
(15)
L mode = T 1L phase T
(16)
L mode =
L + 2M 0 0 0 L M 0 0 0 L M
=
Learth 0 0 0 Lair1 0 0 0 Lair2
(21)
L p_com = TL m_com T
1
(22)
L p_com =
C0_com + 2Ccom Ccom Ccom Ccom C0_com + 2Ccom Ccom Ccom Ccom C0_com + 2Ccom
(23)
Lcom Mcom Mcom Mcom Lcom Mcom Mcom Mcom Lcom
(24)
3.3. Simplification of the compensation circuit The capacitance in the compensation circuit have the same structure as the shunt reactors, which are frequently applied to compensate the reactive power for extra-long transmission lines [28–30]. As a result, simplification of the shunt reactors structure is implemented for compensation circuit in this paper. The original compensation circuit shown by Fig. 4(a) can be redesigned to a simplified model described by Fig. 4(b). The values of new capacitance CN_com and CP_com are calculated as follows 1
1 sC N_com
=
2 s 2C0_com 1 3 + sCcom sC0_com 3
1 sCP_com
=
CN_com =
1
2 s2C0_com
sC0_com
1 3 + sCcom sC0_com
2 C0_com
Ccom
(25)
+ 3C0_com
CP_com = C0_com + 3Ccom
(26)
The deduction of the equations above is illustrated in Appendix. Note that the capacitance between lines is much smaller than the capacitance between the earth and the line, thus Refs. [16,20] proposed
(17)
(18)
where Cm and Lm are the modal capacitance matrix and modal inductance matrix, respectively. The eigenvalues of the characteristic parameter matrices, Eqs. (17) and (18) are classified as air modes (modal circuit where travelling wave propagates between line and line) and earth modes (modal circuit between line and earth). In particular, Cair1, Cair2, Lair1 and Lair2 are defined as air mode capacitances and inductances, respectively; Cearth and Learth are defined as earth mode capacitance and inductance. If the transmission lines are ideally transposed, the parameters of two air modes will be identical, as
Cair1 = Cair2
1
C p_com =
where T denotes the phase-mode transform matrix. Despite different phase-mode transformation matrices, e.g., Clark, Karenbauer, symmetric component transformation, etc. [22–27], the eigenvalues of their diagonalized matrices are identical. The only difference of the diagonalized matrices is the order of arrangement. It is worth mentioning that the inductance and capacitance of the air mode and the earth mode is the same as those of the positive sequence and zero sequence due to the uniqueness of the eigenvalues of their diagonalized matrices. In addition, all the diagonalized matrices can be rewritten as
C0 0 0 Cearth 0 0 = 0 C0 + 3C 0 = Cair1 0 0 0 0 C0 + 3C 0 0 Cair2
C p_com = TC m_com T
where Cp_com and Lp_com are the compensation circuits parameters in phase system as
where uμ and iμ are the voltage and current of phase μ (μ = A, B and C), respectively; C0 are the capacitance between line and earth in p.u.; C and M are the capacitance and mutual inductance between line and line in p.u., respectively; L is the line inductance. The voltage and current in any phase are related to those in the other two phases due to the electromagnetic coupling between three phases, so the inductance and capacitance matrices are both dense matrices, which makes the two-port networks theory inappropriate. To simplify the analysis of multiconductor transmission lines, the phasemode transformation is applied to decouple multiconductor transmission lines into m isolated two-conductor lines described by m separated equations. The phase-mode transformation is indeed a kind of the matrix diagonalization, while the modal inductance matrix Lmode and capacitance matrix Cmode are calculated by
C mode
(20)
Lair1 = Lair2
(9)
Fig. 4. Simplification of the capacitance in the compensation circuit: (a) original compensation circuit; (b) simplified compensation circuit.
(19) 5
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=
N 1
n= 0
µ=
1 N
N 1
(u tuned [n])2
utuned [n] u natural [n] n=0
(u natural [n]) 2 n= 0
(27)
N 1
(u natural [n]
utuned [n]) 2
(28)
n=0
where utuned(x) and unatural(x) are the voltage magnitude distribution along lines of tuned HWTL and natural HWTL, respectively; N is the total number of voltage magnitude distribution; ρ measures the similarity between two voltage distributions, while μ can measure the difference in value of two kinds of voltage distributions.
Fig. 5. The compensation circuit without capacitance between lines.
4. Case studies 4.1. Simulation model In theory, the compensation circuits formed by cascading segments should be scattered along lines to increase the similarity between the natural HWTL and tuned HWTL. However, their implementation along lines may induce extra costs for station construction and equipment maintenance [20]. On the other hand, the implementation of compensation circuits at both ends of lines may cause a similarity degradation of the natural HWTL and tuned HWTL, but the economic costs can be reduced greatly. Therefore, this paper adopts the latter one. The simulations are performed using PSCAD/EMTDC 4.5 program, which is a general-purpose time domain simulation tool for studying transient behaviour of electrical networks. The simulation is executed in a personal computer (PC) with an IntelR Core (TM) i3 CPU at 2.52 GHz and 3 GB of RAM. Notice that the primary line constants are essential for the frequency wavelength, thus the wavelengths of earth mode and air mode differ greatly due to the considerable difference between the primary line constants of the earth mode and those of air mode. Moreover, term HWTL is used to indicate the half-wavelength of the air mode in the previous study. The wavelength λ of the 1000 kV lines at fundamental frequency can be calculated from the positive parameters in Table 1 by
Fig. 6. The flowchart of the parameter calculation of the compensation circuits.
the compensation circuit without capacitance between the lines, as depicted in Fig. 5. In this paper, the differences between the scheme without capacitance between lines and the scheme with capacitance between lines are compared in Section 4. Lastly, the flowchart of parameter calculation of the compensation circuits is demonstrated in Fig. 6.
=
2
= 1
2 1 2
2L
1 C1
+
2C 2 (R 2 1 1
+
2L 2 ) 1
= 5882 km (29)
Thus, the total length of transmission lines of natural HWTL is λ/ 2 = 2941 km. The schematic diagram of the simulation model, which parameters are taken from Ref. [30], is depicted in Fig. 7. Here, the positive sequence and zero sequence of transmission lines are tabulated in Tables 1 and 2, respectively. Assume the electrical length that need to be compensated is 400 km, the compensation circuit parameters in phase system are calculated by Eq. (25) and Eq. (26), as provided in Table 3. To validate the equalization effect of different compensation circuits, three schemes which are applied as follows
3.4. Evaluation of equalization results The equalization results of compensation circuit can be evaluated by the voltage distribution along lines, which includes two statistical indices, e.g., distribution correlation and voltage difference. The distribution correlation aims to estimate the linear relationship of the voltage distributions along lines of the tuned HWTL and natural HWTL, which can describe the similarity of the variation trend of voltage distribution along lines. On the other hand, the voltage difference aims to evaluate the voltage magnitude difference of the tuned HWTL and the natural HWTL. Moreover, the distribution correlation ρ, which is the correlation coefficient of the correlation analysis, and voltage difference μ, which is the variance [7], are calculated by
Scheme 1: Ten Π segments with the structure of capacitance in Fig. 4 at each end of transmission lines; Scheme 2: One Π segment with the structure of capacitance in Fig. 4
Fig. 7. Schematic diagram of the simulation model. 6
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Table 2 Zero sequence parameters of 1000 kV transmission lines. Resistance R0 (Ω/km)
Inductance L0 (H/km)
Capacitance C0 (F/km)
Fundamental frequency f (Hz)
2.5134 × 10−1
2.6921 × 10−3
8.8036 × 10−9
50
Table 3 Compensation circuit parameters. Lcom (H)
Mcom (H)
C0_com (F)
Ccom (F)
5.8306 × 10−1
2.4689 × 10−1
4.8413 × 10−6
6.5991 × 10−7
at each end of transmission lines; Scheme 3: Ten Π segments with the structure of capacitance in Fig. 5 at each end of transmission lines. Fig. 9. Voltage distribution of normal operation with no-load.
4.2. Normal operation
Table 4 Equalization results of normal operation with natural power load.
Some of the advantages of HWTL are that HWTL don’t require the reactive power compensation or no overvoltage restriction for long lines without loads, are obvious in the condition of the normal operation. Thus, the voltage distribution in the normal operation is particularly suitable to verify the similarity of voltage distribution of tuned HWTL and natural HWTL. Set the electrical length as the horizontal axis to compare the difference of tuned HWTL and natural HWTL. The natural power load and non-load are the classical cases to demonstrate the voltage distribution along HWTL lines, as shown in Figs. 8 and 9, respectively. Meanwhile, the distribution correlations and voltage differences of different schemes are presented in Tables 4 and 5, which verify the similarity to natural HWTL. Theoretically, the voltage magnitudes of the natural HWTL at both ends should be identical if the transmission lines are lossless. However, the resistance of the actual transmission lines cannot be neglected, such that the voltage at the receiving end is usually lower than that at the sending end. The voltage distribution of the natural HWTL with natural power load is close to a straight line connecting the sending end and the receiving end. Meanwhile, it can be clearly seen from Fig. 8 that the voltage differences of natural HWTL and tuned HWTLs of scheme 1 and 2 are quite small. Moreover, the variation trend of the voltage distributions along lines of these two schemes are almost identical under natural power. However, the equalization result of scheme 3 is
Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
1.00 0.69
1.00 0.62
0.86 3.29
Table 5 Equalization results of normal operation with no-load. Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
1.00 5.96
1.00 5.97
0.97 61.5
considerably different from the natural HWTL in the voltage distribution, especially in variation trend. Indeed, the absence of the capacitance between lines in scheme 3 induces an imbalance between the inductance and capacitance of transmission lines, which causes the variation trend of voltage distribution along lines to be similar to an attenuated sine curve. In the non-load case, the voltage at the middle point of transmission lines is nearly zero and the voltage at any point of the transmission lines is lower than that at the sending end, which is one of the main advantages of HWTL. Additionally, the voltage distribution of natural HWTL and tuned HWTL of scheme 1 and 2 matches each other at most region, but the tuned HWTL of scheme 3 contains a different voltage minimum location from natural HWTL. The missing parts of capacitance between lines induce differences of tuned HWTL of scheme 3 and natural HWTL, so the capacitance between lines are necessary to equalize the voltage distribution along lines of natural HWTL. Comprehensive simulations in the presence of different loads are carried out to fully compare the equalization results of different schemes, which are only summarized in Table 6 due to the page limits. The voltage at the middle point of the transmission lines rises as load increases. If the load exceeds the natural power, the voltage at the middle point of transmission lines will be larger than that at the sending point, which is totally different from the normal long lines [16]. It can be concluded from Table 6 that the schemes with capacitance between lines have a better equalization results than those without capacitance between lines regardless of the variation of loads. Due to the imbalance of the mutual inductance and the capacitance between phases, the voltage distributions of all the cases with loads are presented as similar to the attenuated sine curve, which significantly reduces the similarity to the natural HWTL. Therefore, the correlation analysis and the
Fig. 8. Voltage distribution of normal operation with natural power load. 7
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Table 6 Equalization results of different loads.
Table 7 Equalization results of single-phase fault.
Load
Index
Scheme 1
Scheme 2
Scheme 3
Index
Scheme 1
Scheme 2
Scheme 3
20% natural power
Ρ Μ
1.00 4.68
1.00 4.69
0.95 61.5
ρ μ
0.99 7.08
0.99 6.64
0.99 6.63
50% natural power
Ρ μ
1.00 1.51
1.00 1.56
0.92 30.7
80% natural power
ρ μ
1.00 0.90
1.00 0.89
0.89 20.4
120% natural power
ρ μ
1.00 1.71
1.00 1.71
0.85 13.1
150% natural power
ρ μ
1.00 2.55
1.00 2.54
0.81 26.5
Table 8 Equalization results of different fault distances. Fault distance
Index
Scheme 1
Scheme 2
Scheme 3
600 km
ρ μ
0.99 6.87
0.99 6.69
0.99 6.72
1000 km
ρ μ
1.00 5.89
1.00 5.43
1.00 5.50
variance applied to calculate the distribution correlation and the voltage difference of scheme 3 are much larger than that of the other schemes.
1400 km
ρ μ
1.00 4.70
1.00 4.71
1.00 4.71
1700 km
ρ μ
0.99 4.99
0.99 4.79
0.99 4.80
4.3. Single-phase fault
2000 km
ρ μ
1.00 5.89
1.00 5.44
1.00 5.54
The transmission lines can operate continuously in several periods of power frequency after the occurrence of the single-phase fault. Hence, the voltage distribution of the faulted phase should be analyzed to compare the equalization results of different schemes. A phase-to-ground (AG) fault with 10 Ω fault resistance and 2595 km away from bus M in natural HWTL is investigated, which represents 2595 km of electrical length in tuned HWTL. The voltage distribution and equalization results are depicted in Fig. 10 and Table 7, respectively. One can observe that the voltage distributions along lines of different tuned HWTL are all similar to the natural HWTL. However, the voltage along lines of natural HWTL is a little bit smaller than that of the tuned HWTL, which is resulted from the resistance of the transmission lines and lossless compensation circuits. Although the resistance of transmission lines is always neglected due to its small value compared with the reactance, the resistance can somehow reduce the overvoltage at the transmission lines. More cases of the single-phase fault are simulated to fully compare the equalization results of different schemes. Due to the page limits, only some of the simulation results are tabulated in Table 8. The fault distance in Table 8 represents the electrical distance away from bus M. All the tested compensation circuit structures cause little difference among them in the single-phase fault cases, which have similar voltage distribution among lines of the natural HWTL. The fault may induce
severe overvoltage on the transmission lines. Moreover, the peak value of the overvoltage is at the point that is a quarter wavelength away from the fault, due to the characteristics of the input impedance of transmission lines. The overvoltage induced by the fault is severe, but the transmission lines sustain the overvoltage for only a few periods. From the viewpoint of overvoltage restriction, the surge arrester and high-speed grounding switch [31], which have been reported to restrict the overvoltage for ultra-high voltage AC transmission lines, may be useful methods for HWTL. The transients induced by faults can transmit along transmission lines to the substation, which is applicable for the protection relay. Thus, the transients which transmit through the compensation circuits should be analyzed to detect the distortion between the compensation circuit and transmission lines. The instantaneous voltage contains the transients induced by the AG fault 400 km away from bus N in terms of electrical length, which is acquired at bus N and illustrated in Fig. 11. Although the cut-off frequency of compensation circuits can be increased by expanding the segment number, the high- frequency components can hardly pass the compensation circuit and transmit to the transformer, as demonstrated in Fig. 11. The transients obtained at bus N contain only a part of the transients at low frequency, which are inadequate to fully represent the fault waveform. Besides, there is no
Fig. 10. Voltage distribution of single-phase fault.
Fig. 11. The instantaneous voltage acquired at bus N. 8
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Table 9 Equalization results of open-phase operation. Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
0.98 1.10
0.97 1.10
0.90 27.81
and different lowest voltage location on transmission lines, even though its voltage distribution is also similar as a sine wave. The transmission lines are assumed as ideal-transposed in this manuscript, but the non-ideal transposition is almost inevitable during the transmission line construction. For such extremely long transmission lines, the non-homogeneous can induce significant inequality of the impedance relative to the ground and mutual impedance between phases, which causes the unbalance of the three phases, or even the obvious variation of power frequency wavelength. Therefore, the compensation circuit can be regarded as an invaluable method to equalize the impedance of three phases of the whole lines by adjusting the parameters of the compensation circuits. Even though the inequality is too large to be completely compensated, the compensation circuits can reduce the inequality to some extent. Compare with the other schemes of reducing the actual length of HWTL, the proposed compensation circuit presents better similarity to the transmission lines. In Ref. [16], only the air modes of the compensation circuit are considered due to its tabulated parameters of the two-port circuit. While the compensation circuit the proposed by this manuscript, which considers the compensation between phases, can achieve a better equalization result. The scheme of high frequency halfwavelength transmission lines proposed in Ref. [18] requires the ACDC-AC converters for rectification of the power frequency to a higher frequency, which is similar to the HVDC power transmission. Compared with the high frequency half-wavelength transmission lines, the tuned half-wavelength transmission lines do not need the large capacity electronic devices, which is a large economic reduction. In Table 10, the compensation circuit without capacitance between phases presented in Ref. [16], and the increasement of power frequency illustrated in [18], which decrease the actual length of the HWTL, are compared with the compensation circuit proposed in this paper.
Fig. 12. Voltage distribution of open-phase operation.
resistance in the compensation circuit to restrain the transient voltage, so the decay of oscillation induced by compensation circuit is much slower than the transmission lines, which have frequency- dependent resistance to consume the energy of high-frequency components. As the analysis at Section 2, the cut-off frequency of the compensation circuit is increasing with the rising of the number of segments, but the segment number has to reach several hundreds to achieve the cut-off frequency up to about 1 MHz, which can pass the fault induced traveling wave. Furthermore, the more segments are applied, the more distorted the transients will be. Therefore, a simple expansion of segment number just for the increase of passband is unnecessary. The compensation circuit is able to pass the power frequency component as transmission lines regardless its block or distortion of the high frequency components induced by the fault. The monitoring system based on the power frequency cannot be affected by the compensation circuit. On the other side, the fault induced transients are complete at the end of actual transmission lines, which can be acquired by the monitoring system between the compensation circuit and transmission lines. Moreover, implementation of the compensation circuit is similar to the series and shunt FACTS devices, which have never adversely influenced the operation of power system or monitoring system.
5. Conclusions The compensation circuit that considers the mutual inductance and capacitance between phases is developed from the mode system in this paper, which main findings/contributions can be summarized as follows:
4.4. Open-phase operation
(1) The analysis of frequency responses of different compensation circuits show that T and L structure compensation circuits may amplify the magnitude of some frequency components. Moreover, time delays of all kinds of compensation circuits are the same due to the identical numbers of series inductance and shunt capacitance. Based on the aforementioned characteristics, Π structure compensation circuit is more suitable for electrical length increasement than T and L structure compensation circuits. (2) In order to be similar to the natural HWTL, the capacitance between phases should be considered in the compensation circuit. Although the compensation circuit without the capacitance between phases can partly equalize the transmission lines, the compensation circuit
Assume the breakers of faulted phase have been opened and the HWTL keeps the open-phase operation to wait for the auto-reclosing, the voltage distribution and equalization results along the phase lines that are still on operation are illustrated in Fig. 12 and Table 9, respectively. The open-phase operation induces that the voltage distribution presents a curve similar as a sine wave. Clearly, the tuned HWTLs of scheme 1 and scheme 3 have almost the same voltage distribution with the natural HWTL, while they have tiny voltage difference. However, the equalization result of the tuned HWTL without the capacitance between lines is not satisfactory due to the larger voltage differences Table 10 Comparation of different methods for decreasing the actual length of HWTL. Method
Compared to the proposed compensation circuit
Compensation circuit without capacitance between phases in Ref. [16] Increasement of power frequency in [18]
The similarity to the transmission lines is inferior to the proposed compensation circuit Frequency conversion requires electronic devices with high economic costs just as HVDC
9
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including the capacitance between phases can significantly improve the equalization results by the simulation of normal operation, single-phase fault and open operation. In addition, the tuned HWTL is based on the theory and characteristics of natural HWTL, so the different voltage distribution of them may induce further technical problems. (3) The cut-off frequency of the compensation circuit can be increased by expanding the number of segments in compensation circuit, in which some high frequency components can pass through. However, a large quantity of transients, such as the traveling wave
(above 100 kHz) which needs hundreds or thousands of segments is hard to achieve. Furthermore, the expanding number of segments can induce severe signal distortion, which is unsatisfactory to increase the cut-off frequency in practice. Declaration of Competing Interest I can hereby confirm there is no interests conflict of the work to anyone.
Appendix Assume the two circuits, demonstrated in Figs. A1 and A2, are equivalent, Eq. (1) can be obtained based on Kirchhoff voltage laws. n
n
Ik1 = k= 2
Yk1(Uk
U 1) = 0
(1)
k=2
The voltage of U1 can be calculated by n
Uk =
Yk1Uk /Y
(2)
k=2
where n
Y =
Yk1
(3)
k=2
To any point i, the voltage of these two circuits can be written as
Fig. A1. Netorks with five nodes.
Fig. A2. Netorks with four nodes. 10
Electrical Power and Energy Systems 115 (2020) 105520
P. Cao, et al. n
Yi1(Ui
U 1) =
Y ik (Ui
Uk ) = 0
k=2 k i
(4)
Based on Eqs. (1) and (4) n k=2
Yi1(Ui
Yk1Uk / Y ) = Yi1(Ui = Yi1
n k=2 k i
n k=2
Yk1(Ui
Yk1
n k=2
Yk1Uk )/ Y
Uk )/ Y
(5)
Hence, Eq. (4) can be rewritten as n
n
Yi1
Yk1(Ui
Uk )/ Y =
k=2 k i
Yk1 (Ui
Uk )
k=2
(6)
Due to the equivalence of Eq. (6) at any voltage, the admittance of these two circuits can be written by n
Y ij = Yi1Yj1/Y
= Yi1Yj1/
Yk1
(7)
k=2
Assume Y23 = Y24 = Y25 = Y0_com , Y34 = Y45 = Y35 = Ycom , Eq. (8) can be obtained from Eq. (7) as
Y12 =
2 Y0_com + 3Y0_com Ycom
(8)
Ycom
Y13 = Y14 = Y15 = Y0_com + 3Ycom If Y0_com = sC0_com and Ycom = sCcom, Eq. (23) can be obtained by
CN_com =
2 C0_com
Ccom
+ 3C0_com
CP_com = C0_com + 3Ccom
(10)
[15] Aredes M, Dias R. FACTS for tapping and power flow control in half-wavelength transmission lines. IEEE Trans Ind Electron 2012;59(10):3669–79. [16] Prabhakara FS, Parthasarathy K, Ramachandra HN. Performance of tuned halfwave-length power transmission lines. IEEE Trans Power Apparat Syst 1969;88(12):1795–802. [17] Ortega JS, Tavares MC. New perspectives about AC link based on half-wavelength properties for bulk power transmission with flexible distance. IET Gener Transm Distrib 2018;12(12):3005–12. [18] Wang Y, Xu W, Li Y, Hao T. High-frequency, half-wavelength power transmission scheme. IEEE Trans Power Delivery 2017;32(1):279–84. [19] Jiao C, Qi L, Cui X. Compensation technology for electrical length of half-wavelength ac power transmission lines. Power Syst Technol 2011;35(9):17–21. [20] Cui X. Chained number of lumped-circuits for physical analogy of the lossless transmission lines. Proc CSEE 2017;37(9):2561–70. [21] Winder S. Analog and digital filter design. 2nd ed. Boston, USA: Newnes; 2002. p. 46–7. [22] Lopes FV. Settings-free traveling-wave-based earth fault location using unsynchronized two-terminal data. IEEE Trans Power Delivery 2016;31(5):2296–8. [23] Caballero PT, Costa ECM, Kurokawa S. Modal decoupling of overhead transmission lines using real and constant matrices: influence of the line length. Int J Electr Power Energy Syst 2017;92:202–11. [24] Zhang Y, Liang J, Yun ZH, Dong XM. A new fault-location algorithm for seriescompensated double-circuit transmission lines based on the distributed parameter model. IEEE Trans Power Delivery 2017;32(6):2398–407. [25] Yao W, Jiang L, Wen JY, Wu QH, Cheng SJ. Wide-area damping controller of FACTS devices for inter-area oscillations considering communication time delays. IEEE Trans Power Syst 2014;29(1):318–29. [26] Liu J, Wen JY, Yao W, Long Y. Solution to short-term frequency response of wind farms by using energy storage systems. IET Renew Power Gener 2016;10(5):669–78. [27] Luo X, Huang C, Jiang Y, Guo S. An adaptive three-phase reclosure scheme for shunt reactor-compensated transmission lines based on the change of current spectrum. Electr Power Syst Res 2018;158:184–94. [28] Oleksyuk BV, Tulsky VN, Palis S. Magnetically controlled shunt reactors as sources of current and voltage harmonics. IEEE Trans Power Delivery 2017;33(4):1818–24. [29] Tümaya M, Demirdelen T, Bal S, Kayaalp R, Doğru B, Aksoy M. A review of magnetically controlled shunt reactor for power quality improvement with renewable energy applications. Renew Sustain Energy Rev 2017;77:215–28. [30] Fabián RG, Tavares MC. Faulted phase selection for half-wavelength power transmission lines. IEEE Trans Power Delivery 2017;33(2):992–1001. [31] Nakamura A, Okamoto H, Cao X. Introduction to 1000kv transmission technologies conducted by tokyo electric power company. Power Syst Technol 2005;29(6):1–7.
References [1] Yang B, Zhang XS, Yu T, Shu HC, Fang ZH. Grouped grey wolf optimizer for maximum power point tracking of doubly-fed induction generator based wind turbine. Energy Convers Manage 2017;133:427–43. [2] Yang B, Jiang L, Wang L, Yao W, Wu QH. Nonlinear maximum power point tracking control and modal analysis of DFIG based wind turbine. Int J Electr Power Energy Syst 2016;49:429–36. [3] Yang B, Yu T, Shu HC, Zhang YM, Chen J, Sang YY, et al. Passivity-based slidingmode control design for optimal power extraction of a PMSG based variable speed wind turbine. Renew Energy 2018;119:577–89. [4] Dash PK, Patnaik RK, Mishra SP. Adaptive fractional integral terminal sliding mode power control of UPFC in DFIG wind farm penetrated multimachine power system. Prot Control Mod Power Syst 2018;3(3):79–92. [5] Zhou M, Xiang W, Zuo W, Lin W, Wen J. A novel HVDC circuit breaker for HVDC application. Int J Electr Power Energy Syst 2019;109:685–95. [6] Santos MLD, Santo SGD. Evaluation of voltage and current profiles and Joule losses for a half-wavelength power transmission line. Int J Electr Power Energy Syst 2015;70(9):39–44. [7] Zou G, Huang Q, Song S, Tong B, Gao H. Novel transient-energy-based directional pilot protection method for HVDC line. Prot Control Mod Power Syst 2017;2(2):159–68. [8] Samorodov G, Kandakov S, Zilberman S, Krasilnikova T, Tavares MC, Machado C, et al. Technical and economic comparison between direct current and half-wavelength transmission systems for very long distances. IET Gener Transm Distrib 2017;11(11):2871–8. [9] Araujo MR, Soares LA, Castro AGA, Drumond WBF, Coelho ALMA. performance evaluation approach of commercial numerical distance relays on little longer than half-wavelength transmission lines. 2018 Simposio Brasileiro de Sistemas Eletricos (SBSE), Niteroi. 2018. p. 1–6. [10] Tang LX, Dong XZ. A travelling wave differential protection scheme for half-wavelength transmission line. Int J Electr Power Energy Syst 2018;99:376–84. [11] Dias O, Tavares MC. Single-phase auto-reclosing mitigation procedure for halfwavelength transmission line. IET Gener Transm Distrib 2017;11(17):4324–31. [12] Lopes FV, Kusel BF, Silva KM. Traveling wave-based fault location on half-wavelength transmission lines. IEEE Lat Am Trans 2016;14(1):248–53. [13] Lopes FV, Kusel BF, Silva KM, Fernandes Jr, Neves WLA. Fault location on transmission lines little longer than half-wavelength. Electr Power Syst Res 2014;114:101–9. [14] Silva EA, Moreira FA, Tavares MC. Energization simulations of a half-wavelength transmission line when subject to three-phase faults-Application to a field test situation. Electr Power Syst Res 2016;138:58–65.
11
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Analysis of electrical length compensation types for tuned half-wavelength transmission lines☆
T
Pulin Caoa, Hongchun Shua, Bo Yanga, , Shengnan Lib, Tingyi Heb, Lei Yangb, Yiming Hana, Tao Yuc ⁎
a
Faculty of Electric Power Engineering, Kunming University of Science and Technology, 650500 Kunming, China Electric Power Research Institute of Yunnan Power Grid Co., Ltd., 650217 Kunming, China c College of Electric Power, South China University of Technology, 510640 Guangzhou, China b
ARTICLE INFO
ABSTRACT
Keywords: Compensation circuit Frequency response Half-wavelength transmission lines (HWTL) Modal-phase transformation Power transmission
The compensation circuits are the core to improve the flexibility of practical application of half-wavelength transmission lines (HWTL), which are restricted to the fixed transmission length due to the constant fundamental frequency. This paper proposes a mode-phase transformation based method to design the parameters of compensation circuits after comparison of frequency responses of three kinds of compensation circuits, e.g., T segments, Π segments and L segments. At first, the compensation circuit parameters are obtained in the modal system that separates the coupling multiconductor system into isolated circuits to avoid the complexity of the three-phase system. Then, the compensation circuit parameters can be transformed into multiconductor system by modal-phase transformation, which accomplishes the compensation circuit parameter design in three-phase system. Lastly, the parameters acquired by the proposed method are compared with the natural HWTL. Comprehensive simulations are undertaken which validate that the proposed method is more suitable for tuning HWTL.
1. Introduction DUE to the carbon dioxide emission restriction of the United Nations Framework Convention on Climate Change [1,2], the worldwide demand of renewable energy, which is an ideal replacement of traditional fossil fuel, is apparently ever-increasing. In general, the renewable sources around load centers are insufficient to sustain the fast development of load centers in countries with continental extensions, such as Brazil, Russia, and China. In fact, the small-scale utilization of the renewable energy, which requires just some simple equipment, can be easily implemented near the low-capacity load such as the microgrid without strict demand of operating conditions [3]. However, the largescale renewable energy base requires strict climatic and geographic conditions, i.e., appropriate wind speed, annually stable sunlight and consistently abundant wind resource [4]. This dilemma always occurs at the countries with continental extensions, in which regional economic growth is notably unbalanced. For example, the Gangsu wind farm is the world’s largest wind farm with a capacity of 6800 MW and is expected to reach a total capacity of 20,000 MW by 2020, is located in
Gangsu province in the undeveloped area of northwest China with very few population and power demand. Nevertheless, Gangsu province is unable to consume all the capacity of this large-scale wind farm, hence most of its wind power has to be transmitted to the load centers located at east coast of China, e.g., Shanghai and Guangzhou, which is about 2600 km away from Gangsu. Therefore, the bulk power transmission over thousands of kilometers is undoubtedly inevitable to overcome the extremely long distance between large-scale power plants and load centers. To remedy the above challenge, ultra-high voltage direct current (UHVDC) systems and ultra-high voltage alternative current (UHVAC) systems have been regarded as a prominent solution for the bulk electric power transmission with extremely long distance in China and Brazil. In general, considerable reactive power compensation and protection scheme improvement is required for UHVAC when the length of transmission line is much smaller than half-wavelength, which has resulted in few applications of UHVAC [5]. Unfortunately, there is no real half-wavelength transmission lines in practice. Although the low capacity half-wavelength transmission lines
This work was supported in part by the National Natural Science Foundations of China under Grant 51807085, 51977102 and 61963020, and in part by the Yunnan Provincial Talents Training Program under Grant KKSY201604015. ⁎ Corresponding author. E-mail address: [email protected] (B. Yang). ☆
https://doi.org/10.1016/j.ijepes.2019.105520 Received 24 January 2019; Received in revised form 21 August 2019; Accepted 27 August 2019 Available online 02 September 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
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(HWTL) are widely applied in the communication engineering domain and the basic concepts of half-wavelength power transmission have been proposed since 1939 [6], the large capacity HWTL with ultra-high voltage are prudently considered in Russia, Brazil and China, which has not been implemented in practice due to the temporarily insufficient power demand and other technical problems such as its constant line length. However, such technique is prominent and worth investigating in future as the power demand is increasing rapidly while more advanced techniques appear. In the past decade, HWTL has gained significant interests around the globe thanks to the technological advancements, especially the improvements of ultra-high voltage transmission line operation and control. Compared to the widely applied high voltage direct current (HVDC) which is a mature technology nowadays for bulk power transmission over extensive geographical region [7], HWTL does not need expensive high-voltage converters, such that the installation costs could be reduced. Ref. [8] concludes that the investment for the single-circuit HWL with reverse phase can be considered 30% (1.15 billion USD) lower than that for HVDC and the total yearly cost is 15% lower on the condition of the power transmission of 6000 MW over 2500 km. In addition, the traditional singleend relay protections are unable to distinguish the remote-end faults from faults on other lines connected to the same bus without auxiliary devices due to the extremely long distance of HWTL. The nonlinear variation of the impedance induces that the short-circuit impedance of HWTL varies between infinity and zero, hence the traditional distance protection that needs to calculate the short-circuit impedance may lose efficiency in HWTL, thus the conventional protection is unable to applied directly for HWTL. Ref. [9] proposed a scheme with five kinds of protection to promptly identify the fault and cover the whole line from single-end in order to promptly identify the fault and cover the whole line by traditional protection. In addition, travelling wave based differential protection scheme and have been applied by literatures [10] for HWTL, which attempt to overcome the adverse effect of extremely long distance that may invalidate the relay protections. In order to promote the success rate of auto-reclosing of HWTL, literature [11] implemented a reactor to ground the power transformers after the fault occurring. Moreover, the long distance of HWTL is also quite inconvenient for the fault detection and search, thus a fault location technique based on travelling wave has been proposed for faults that generates obvious travelling waves [12]. However, the attenuation of high frequency components caused by faults or breaker operations are so severe in the long distance transmission, such that the high frequency components are usually difficult to be detected at bus station. Thus, Ref. [13] proposed a fault location method relying on symmetrical components. In order to achieve the characteristics of half-wavelength, the length of HWTL has to be fixed around 3000 km or 2500 km due to the constant fundamental frequency of 50 Hz or 60 Hz, [14], which therefore restricts the practical application of HWTL. In order to control the power flow of the HWTL, the series and shunt flexible ac transmission systems (FACTS) are applied in HWTL to control the main power flow [15], which demonstrates the feasibility of the FACTS to variate the power flow in HWTL. Based on the communication engineering, work [16] proposed a compensation algorithm to shorten a transmission line than half-wavelength to the tuned HWTL with compensation circuits consisted of inductance and capacitance. Additionally, three kinds of compensation circuit, e.g., T segment, Π segment and distributed shunt capacitors, have been developed, which concluded that the distributed shunt capacitors are less efficient than T segment and Π segment formed compensation circuits. Compared with HVDC, literature [17] analyzed the flexible application of the tuned HWTL, which shows that the transmission efficiency of the tuned HWTL is similar to that of HVDC. Moreover, work [18] attempted to increase the flexibility of HWTL by generating high fundamental frequency from 180 Hz to 360 Hz to make HWTL suitable for the short-distance transmission, in which the length of HWTL can be reduced to less than 1000 km.
Although the frequency conversion significantly decreases the length of HWTL, an AC-AC or an AC-DC-AC conversion at the ends of transmission lines is required to transform high frequency power to 50 Hz or 60 Hz component for traditional fundamental frequency loads, which requires high voltage and large capacity electronic devices as HVDC. Thus, high costs of electronic devices are inevitable in this method, which significantly increases the overall investment. Although the compensation types for tuned half-wavelength transmission lines have been presented in the Ref. [16], the necessary of the capacitance between phases should be further considered. On the other side, the segments of T, Π and L have been depicted in other Ref. [19], but their advantages and the differences for compensation circuits, especially their frequency responses need to be analyzed. Furthermore, the diverse compensation types and their cascades replace the actual transmission lines, of which only the influence at the stable voltage distribution on the transmission lines is studied [20]. Due to the outstanding performance at stable voltage distribution on the transmission lines, a large quantity of cascades of the simple compensation circuit is believed to be an excellent choice to replace the transmission lines. Hence whether a large number of cascades of segments is necessary for the compensation circuit should be studied. The reduction of the actual length of transmission lines is the main advantage of the tuned HWTL. Being similar to the natural HWTL is able to restrict the technical problems similar to the natural HWTL. If the compensation circuit changes the unique characteristics of HWTL, more problems, such as voltage distribution along lines and phase variation, have to be considered, which further increase the complexity of the technical problems. Thus, the similarity of the compensation circuit and transmission lines is critically necessary. In this paper, the frequency responses of compensation circuits formed by T segment, Π segment and L segment are analyzed to present the frequency response variations resulted from the different segment numbers and structures. Moreover, the compensation circuits regarded as two-port circuits are firstly designed in the modal system. Then the compensation circuit parameters are transformed into three-phase system for compensation circuit design. Lastly, comprehensive case studies are carried out to verify the effectiveness of the proposed compensation-circuits in comparison to the traditional compensation circuits. The main contributions of this manuscript can be summarized as: (1) Considering the different parameters of air mode and earth mode, this work proposes a compensation parameter calculation scheme, which calculates the parameters of inductance and capacitance in mode system and then transforms the parameters to the phase mode to complete the parameters of the compensation circuit design. The mutual inductance and capacitance between phases are fully taken into account in this design scheme, which makes the designed compensation circuit more similar to the transmission lines than the other compensation circuits. (2) Due to the outstanding performance at stable voltage distribution at the transmission lines, a large quantity of cascades of the simple compensation circuit is believed to be an excellent choice to replace the transmission lines. The transient waveform of the tuned halfwavelength transmission lines is analyzed in this manuscript to verify that the number of the cascades of the compensation circuit should be restricted to avoid the oscillations induced by compensation circuits. (3) Based on the frequency response analysis, the structure of Π segment is verified to be a better compensation circuit unit than the other two that may amplify some frequency components. 2. Equalization of lossless two-port transmission lines In order to accurately represent the transmission lines by lumped circuits, the lossless model, Π and T models, are often adopted. In addition, they are represented by an infinite series of the L model in 2
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Table 1 Positive sequence parameters of 1000 kv transmission lines. Resistance R1 (Ω/km)
Inductance L1 (H/km)
Capacitance C1 (F/km)
Fundamental frequency f (Hz)
1.0324 × 10−2
8.4042 × 10−4
1.3752 × 10−8
50
unnecessary. Although the frequency response at fundamental frequency is the core of compensation circuit for power transmission, the frequency responses of other frequency components, especially the poles and zeros that may cause relay protection failure and overvoltage, should also be studied. The lossless compensation circuit for HWTL illustrated in Figs. 1 and 2 have the same structure of the low pass constant k filter which consists of a ladder network of identical sections of passive circuit components [21]. Hence, the theory of passive analog filter can be performed to calculate and analyze the frequency responses of the compensation circuit. The positive sequence parameters of the transmission lines which are implemented in 1000 kV ultra-high voltage Jingdongnan-NanyangJingmen lines in National Power Grid in China are tabulated in Table 1. The propagation constant γ of transmission lines can be calculated by [10]
Fig. 1. Compensation circuit models (a) Π model, (b) T model, (c) L model.
transient analysis. Thus, three kinds of lumped circuit models can be adopted for equalization of the transmission lines to increase the electrical length. In Fig. 1, l is the length of transmission lines; Lline and Cline are the line inductance and capacitance in per unit (p.u.), respectively. Then, each aforementioned equivalent model can form composite section by cascading n Π, T and L segments, which increase the similarity of the compensation circuit and distribution model of transmission lines, as demonstrated in Fig. 2. In the extreme case, the compensation circuit depicted in Fig. 2 can be regarded as transmission lines if the inductance and capacitance are infinitesimal. It composes an infinite series two-port circuit which are the fundamental hypothesis of lossless telegrapher equations. Since the segments number cannot be infinite in practice, the frequency responses of the compensation circuit are quite different from that of transmission lines. In order to properly equalize the normal power transmission and electromagnetic transients in the transmission lines, the frequency responses of compensation circuit should be much similar to that of transmission lines. However, it can only be obtained if the compensation circuit has the same frequency-dependent parameter to transmission lines, which is usually impossible due to the different structures and materials of transmission lines and lumped circuit. On the other hand, the compensation circuit is implemented to increase the electrical length at fundamental frequency to half-wavelength. It requires the magnitude attenuation at fundamental frequency to be as small as possible and the phase shifting to be the same to transmission lines. Hence, the accurate equalization of transmission lines spectrum is
=
+j =
(1)
(R1 + j L1)(G1 + j C1)
where α and β are the attenuation constant and phase constant, respectively; R1, L1, G1 and C1 are the resistance, inductance, conductance and capacitance per unit of the transmission lines. The attenuation constant and phase constant can be obtained by the equation above, as
=
1 [R1 G1 2
=
1 [ 2
+
(R12 +
2L 2 )(G 2 1 1
+
2C 2 ) ] 1
R1 G1 +
(R12 +
2L 2 )(G 2 1 1
+
2C 2 ) ] 1
2L
2L C 1 1
1 C1
(2) (3)
The conductance of the transmission line is small when compared to other parameters, and is neglected in these equations. The attenuation constant α, phase constant β and characteristic impedance Zc at fundamental frequency can be calculated by
=
1 [ 2
2L
=
1 [ 2
2L C 1 1
Zc =
1 C1
+
(R12 +
+
(R12 +
2L 2 ) 1
2L 2 ) 1
2C 2 ) ] 1
2C 2 ] 1
R1 + j L1 = 247.2 + j4.832 j C1
= 2.088 × 10
5
NP/km
= 1.068 × 10 - 3 rad/km
(4) (5) (6)
where ω = 2πf. The compensation circuit is aimed to increase the electrical length for transmission lines near half-wavelength, hence the extended electrical length is not supposed to be too large. Assume the electrical length Len that needs to be compensated is 100 km, the magnitude decay coefficient D and the time delay Td of the transmission lines at fundamental frequency are described by
D = e Len × 100% = 99.8%
Td =
2 f
Len = 340 µs
(7) (8)
Note that this paper selects 100 km compensation circuit as an example to increase the electrical length for transmission lines near halfwavelength, in which the extended electrical length is not supposed to be too large. The compensation circuits are comprised by 1, 2, 5 and 10 segments. Meanwhile, the characteristic impedance Zc is chosen as the
Fig. 2. Composite sections of compensation circuits (a) Π segments, (b) T segments, (c) L segments. 3
P. Cao, et al.
Electrical Power and Energy Systems 115 (2020) 105520
Fig. 3. Magnitude responses of compensation circuits (a) Π segments, (b) T segments, (c) L segments; phase shifting of compensation circuits: (d) Π segments, (e) T segments, (f) L segments.
Fig. 3. (continued)
by increasing the segment number. However, it may cause severer signal distortion, which degrades the reliability of protection relay or fault detection equipment in the substations.
termination. The output magnitude and the delay of frequency components at termination are given in Fig. 3. It can be observed that the decay magnitude of fundamental frequency of each compensation circuit is almost zero. Meanwhile, the time delay at fundamental frequency of each compensation circuit is 340 μs, which are the same to the transmission lines. Therefore, all the compensation circuits are suitable to replace the transmission lines for power transmission at fundamental frequency. However, the magnitude and the number of the frequency components, which can be amplified by compensation circuits to induce overvoltage, increase in both L and T compensation circuits as the segments number grows. Therefore, the Π segments formed compensation circuit, which does not increase any frequency component magnitude is more suitable to equalize the transmission lines. Furthermore, the passband of compensation circuit can be extended
3. Compensation circuits for multiconductor transmission line 3.1. Phase-mode transformation for lossless multiconductor transmission lines Although the compensation circuit can be easily designed for the two-port transmission lines, the power grid for bulk AC power transmission is multiconductor transmission lines with three parallel conductors, which cannot be analyzed as simple two-port transmission lines. The relationship of primary line constants, voltage and current in lossless three-phase transmission lines can be written as follows 4
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iphase x
uphase x
= C phase
= L phase
uphase t
3.2. Parameter calculation in phase lines
iphase
(10)
t
The compensation circuit parameters are calculated to compensate the electrical length of different modal circuits. Then, all the parameters in modal circuits should be transformed to phase conductors to accomplish the compensation circuit design. The inductance Lm_com and capacitance Cm_com of the compensation circuit in the modal circuits can be transformed to phase system by
where
iphase = [ iA iB iC ]T
(11)
uphase = [uA uB uC ]T
(12)
C phase =
C0 + 2C C C C C0 + 2C C C C C0 + 2C
(13)
L M M L phase = M L M M M L
(14)
C mode = T 1C phase T
(15)
L mode = T 1L phase T
(16)
L mode =
L + 2M 0 0 0 L M 0 0 0 L M
=
Learth 0 0 0 Lair1 0 0 0 Lair2
(21)
L p_com = TL m_com T
1
(22)
L p_com =
C0_com + 2Ccom Ccom Ccom Ccom C0_com + 2Ccom Ccom Ccom Ccom C0_com + 2Ccom
(23)
Lcom Mcom Mcom Mcom Lcom Mcom Mcom Mcom Lcom
(24)
3.3. Simplification of the compensation circuit The capacitance in the compensation circuit have the same structure as the shunt reactors, which are frequently applied to compensate the reactive power for extra-long transmission lines [28–30]. As a result, simplification of the shunt reactors structure is implemented for compensation circuit in this paper. The original compensation circuit shown by Fig. 4(a) can be redesigned to a simplified model described by Fig. 4(b). The values of new capacitance CN_com and CP_com are calculated as follows 1
1 sC N_com
=
2 s 2C0_com 1 3 + sCcom sC0_com 3
1 sCP_com
=
CN_com =
1
2 s2C0_com
sC0_com
1 3 + sCcom sC0_com
2 C0_com
Ccom
(25)
+ 3C0_com
CP_com = C0_com + 3Ccom
(26)
The deduction of the equations above is illustrated in Appendix. Note that the capacitance between lines is much smaller than the capacitance between the earth and the line, thus Refs. [16,20] proposed
(17)
(18)
where Cm and Lm are the modal capacitance matrix and modal inductance matrix, respectively. The eigenvalues of the characteristic parameter matrices, Eqs. (17) and (18) are classified as air modes (modal circuit where travelling wave propagates between line and line) and earth modes (modal circuit between line and earth). In particular, Cair1, Cair2, Lair1 and Lair2 are defined as air mode capacitances and inductances, respectively; Cearth and Learth are defined as earth mode capacitance and inductance. If the transmission lines are ideally transposed, the parameters of two air modes will be identical, as
Cair1 = Cair2
1
C p_com =
where T denotes the phase-mode transform matrix. Despite different phase-mode transformation matrices, e.g., Clark, Karenbauer, symmetric component transformation, etc. [22–27], the eigenvalues of their diagonalized matrices are identical. The only difference of the diagonalized matrices is the order of arrangement. It is worth mentioning that the inductance and capacitance of the air mode and the earth mode is the same as those of the positive sequence and zero sequence due to the uniqueness of the eigenvalues of their diagonalized matrices. In addition, all the diagonalized matrices can be rewritten as
C0 0 0 Cearth 0 0 = 0 C0 + 3C 0 = Cair1 0 0 0 0 C0 + 3C 0 0 Cair2
C p_com = TC m_com T
where Cp_com and Lp_com are the compensation circuits parameters in phase system as
where uμ and iμ are the voltage and current of phase μ (μ = A, B and C), respectively; C0 are the capacitance between line and earth in p.u.; C and M are the capacitance and mutual inductance between line and line in p.u., respectively; L is the line inductance. The voltage and current in any phase are related to those in the other two phases due to the electromagnetic coupling between three phases, so the inductance and capacitance matrices are both dense matrices, which makes the two-port networks theory inappropriate. To simplify the analysis of multiconductor transmission lines, the phasemode transformation is applied to decouple multiconductor transmission lines into m isolated two-conductor lines described by m separated equations. The phase-mode transformation is indeed a kind of the matrix diagonalization, while the modal inductance matrix Lmode and capacitance matrix Cmode are calculated by
C mode
(20)
Lair1 = Lair2
(9)
Fig. 4. Simplification of the capacitance in the compensation circuit: (a) original compensation circuit; (b) simplified compensation circuit.
(19) 5
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=
N 1
n= 0
µ=
1 N
N 1
(u tuned [n])2
utuned [n] u natural [n] n=0
(u natural [n]) 2 n= 0
(27)
N 1
(u natural [n]
utuned [n]) 2
(28)
n=0
where utuned(x) and unatural(x) are the voltage magnitude distribution along lines of tuned HWTL and natural HWTL, respectively; N is the total number of voltage magnitude distribution; ρ measures the similarity between two voltage distributions, while μ can measure the difference in value of two kinds of voltage distributions.
Fig. 5. The compensation circuit without capacitance between lines.
4. Case studies 4.1. Simulation model In theory, the compensation circuits formed by cascading segments should be scattered along lines to increase the similarity between the natural HWTL and tuned HWTL. However, their implementation along lines may induce extra costs for station construction and equipment maintenance [20]. On the other hand, the implementation of compensation circuits at both ends of lines may cause a similarity degradation of the natural HWTL and tuned HWTL, but the economic costs can be reduced greatly. Therefore, this paper adopts the latter one. The simulations are performed using PSCAD/EMTDC 4.5 program, which is a general-purpose time domain simulation tool for studying transient behaviour of electrical networks. The simulation is executed in a personal computer (PC) with an IntelR Core (TM) i3 CPU at 2.52 GHz and 3 GB of RAM. Notice that the primary line constants are essential for the frequency wavelength, thus the wavelengths of earth mode and air mode differ greatly due to the considerable difference between the primary line constants of the earth mode and those of air mode. Moreover, term HWTL is used to indicate the half-wavelength of the air mode in the previous study. The wavelength λ of the 1000 kV lines at fundamental frequency can be calculated from the positive parameters in Table 1 by
Fig. 6. The flowchart of the parameter calculation of the compensation circuits.
the compensation circuit without capacitance between the lines, as depicted in Fig. 5. In this paper, the differences between the scheme without capacitance between lines and the scheme with capacitance between lines are compared in Section 4. Lastly, the flowchart of parameter calculation of the compensation circuits is demonstrated in Fig. 6.
=
2
= 1
2 1 2
2L
1 C1
+
2C 2 (R 2 1 1
+
2L 2 ) 1
= 5882 km (29)
Thus, the total length of transmission lines of natural HWTL is λ/ 2 = 2941 km. The schematic diagram of the simulation model, which parameters are taken from Ref. [30], is depicted in Fig. 7. Here, the positive sequence and zero sequence of transmission lines are tabulated in Tables 1 and 2, respectively. Assume the electrical length that need to be compensated is 400 km, the compensation circuit parameters in phase system are calculated by Eq. (25) and Eq. (26), as provided in Table 3. To validate the equalization effect of different compensation circuits, three schemes which are applied as follows
3.4. Evaluation of equalization results The equalization results of compensation circuit can be evaluated by the voltage distribution along lines, which includes two statistical indices, e.g., distribution correlation and voltage difference. The distribution correlation aims to estimate the linear relationship of the voltage distributions along lines of the tuned HWTL and natural HWTL, which can describe the similarity of the variation trend of voltage distribution along lines. On the other hand, the voltage difference aims to evaluate the voltage magnitude difference of the tuned HWTL and the natural HWTL. Moreover, the distribution correlation ρ, which is the correlation coefficient of the correlation analysis, and voltage difference μ, which is the variance [7], are calculated by
Scheme 1: Ten Π segments with the structure of capacitance in Fig. 4 at each end of transmission lines; Scheme 2: One Π segment with the structure of capacitance in Fig. 4
Fig. 7. Schematic diagram of the simulation model. 6
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Table 2 Zero sequence parameters of 1000 kV transmission lines. Resistance R0 (Ω/km)
Inductance L0 (H/km)
Capacitance C0 (F/km)
Fundamental frequency f (Hz)
2.5134 × 10−1
2.6921 × 10−3
8.8036 × 10−9
50
Table 3 Compensation circuit parameters. Lcom (H)
Mcom (H)
C0_com (F)
Ccom (F)
5.8306 × 10−1
2.4689 × 10−1
4.8413 × 10−6
6.5991 × 10−7
at each end of transmission lines; Scheme 3: Ten Π segments with the structure of capacitance in Fig. 5 at each end of transmission lines. Fig. 9. Voltage distribution of normal operation with no-load.
4.2. Normal operation
Table 4 Equalization results of normal operation with natural power load.
Some of the advantages of HWTL are that HWTL don’t require the reactive power compensation or no overvoltage restriction for long lines without loads, are obvious in the condition of the normal operation. Thus, the voltage distribution in the normal operation is particularly suitable to verify the similarity of voltage distribution of tuned HWTL and natural HWTL. Set the electrical length as the horizontal axis to compare the difference of tuned HWTL and natural HWTL. The natural power load and non-load are the classical cases to demonstrate the voltage distribution along HWTL lines, as shown in Figs. 8 and 9, respectively. Meanwhile, the distribution correlations and voltage differences of different schemes are presented in Tables 4 and 5, which verify the similarity to natural HWTL. Theoretically, the voltage magnitudes of the natural HWTL at both ends should be identical if the transmission lines are lossless. However, the resistance of the actual transmission lines cannot be neglected, such that the voltage at the receiving end is usually lower than that at the sending end. The voltage distribution of the natural HWTL with natural power load is close to a straight line connecting the sending end and the receiving end. Meanwhile, it can be clearly seen from Fig. 8 that the voltage differences of natural HWTL and tuned HWTLs of scheme 1 and 2 are quite small. Moreover, the variation trend of the voltage distributions along lines of these two schemes are almost identical under natural power. However, the equalization result of scheme 3 is
Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
1.00 0.69
1.00 0.62
0.86 3.29
Table 5 Equalization results of normal operation with no-load. Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
1.00 5.96
1.00 5.97
0.97 61.5
considerably different from the natural HWTL in the voltage distribution, especially in variation trend. Indeed, the absence of the capacitance between lines in scheme 3 induces an imbalance between the inductance and capacitance of transmission lines, which causes the variation trend of voltage distribution along lines to be similar to an attenuated sine curve. In the non-load case, the voltage at the middle point of transmission lines is nearly zero and the voltage at any point of the transmission lines is lower than that at the sending end, which is one of the main advantages of HWTL. Additionally, the voltage distribution of natural HWTL and tuned HWTL of scheme 1 and 2 matches each other at most region, but the tuned HWTL of scheme 3 contains a different voltage minimum location from natural HWTL. The missing parts of capacitance between lines induce differences of tuned HWTL of scheme 3 and natural HWTL, so the capacitance between lines are necessary to equalize the voltage distribution along lines of natural HWTL. Comprehensive simulations in the presence of different loads are carried out to fully compare the equalization results of different schemes, which are only summarized in Table 6 due to the page limits. The voltage at the middle point of the transmission lines rises as load increases. If the load exceeds the natural power, the voltage at the middle point of transmission lines will be larger than that at the sending point, which is totally different from the normal long lines [16]. It can be concluded from Table 6 that the schemes with capacitance between lines have a better equalization results than those without capacitance between lines regardless of the variation of loads. Due to the imbalance of the mutual inductance and the capacitance between phases, the voltage distributions of all the cases with loads are presented as similar to the attenuated sine curve, which significantly reduces the similarity to the natural HWTL. Therefore, the correlation analysis and the
Fig. 8. Voltage distribution of normal operation with natural power load. 7
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Table 6 Equalization results of different loads.
Table 7 Equalization results of single-phase fault.
Load
Index
Scheme 1
Scheme 2
Scheme 3
Index
Scheme 1
Scheme 2
Scheme 3
20% natural power
Ρ Μ
1.00 4.68
1.00 4.69
0.95 61.5
ρ μ
0.99 7.08
0.99 6.64
0.99 6.63
50% natural power
Ρ μ
1.00 1.51
1.00 1.56
0.92 30.7
80% natural power
ρ μ
1.00 0.90
1.00 0.89
0.89 20.4
120% natural power
ρ μ
1.00 1.71
1.00 1.71
0.85 13.1
150% natural power
ρ μ
1.00 2.55
1.00 2.54
0.81 26.5
Table 8 Equalization results of different fault distances. Fault distance
Index
Scheme 1
Scheme 2
Scheme 3
600 km
ρ μ
0.99 6.87
0.99 6.69
0.99 6.72
1000 km
ρ μ
1.00 5.89
1.00 5.43
1.00 5.50
variance applied to calculate the distribution correlation and the voltage difference of scheme 3 are much larger than that of the other schemes.
1400 km
ρ μ
1.00 4.70
1.00 4.71
1.00 4.71
1700 km
ρ μ
0.99 4.99
0.99 4.79
0.99 4.80
4.3. Single-phase fault
2000 km
ρ μ
1.00 5.89
1.00 5.44
1.00 5.54
The transmission lines can operate continuously in several periods of power frequency after the occurrence of the single-phase fault. Hence, the voltage distribution of the faulted phase should be analyzed to compare the equalization results of different schemes. A phase-to-ground (AG) fault with 10 Ω fault resistance and 2595 km away from bus M in natural HWTL is investigated, which represents 2595 km of electrical length in tuned HWTL. The voltage distribution and equalization results are depicted in Fig. 10 and Table 7, respectively. One can observe that the voltage distributions along lines of different tuned HWTL are all similar to the natural HWTL. However, the voltage along lines of natural HWTL is a little bit smaller than that of the tuned HWTL, which is resulted from the resistance of the transmission lines and lossless compensation circuits. Although the resistance of transmission lines is always neglected due to its small value compared with the reactance, the resistance can somehow reduce the overvoltage at the transmission lines. More cases of the single-phase fault are simulated to fully compare the equalization results of different schemes. Due to the page limits, only some of the simulation results are tabulated in Table 8. The fault distance in Table 8 represents the electrical distance away from bus M. All the tested compensation circuit structures cause little difference among them in the single-phase fault cases, which have similar voltage distribution among lines of the natural HWTL. The fault may induce
severe overvoltage on the transmission lines. Moreover, the peak value of the overvoltage is at the point that is a quarter wavelength away from the fault, due to the characteristics of the input impedance of transmission lines. The overvoltage induced by the fault is severe, but the transmission lines sustain the overvoltage for only a few periods. From the viewpoint of overvoltage restriction, the surge arrester and high-speed grounding switch [31], which have been reported to restrict the overvoltage for ultra-high voltage AC transmission lines, may be useful methods for HWTL. The transients induced by faults can transmit along transmission lines to the substation, which is applicable for the protection relay. Thus, the transients which transmit through the compensation circuits should be analyzed to detect the distortion between the compensation circuit and transmission lines. The instantaneous voltage contains the transients induced by the AG fault 400 km away from bus N in terms of electrical length, which is acquired at bus N and illustrated in Fig. 11. Although the cut-off frequency of compensation circuits can be increased by expanding the segment number, the high- frequency components can hardly pass the compensation circuit and transmit to the transformer, as demonstrated in Fig. 11. The transients obtained at bus N contain only a part of the transients at low frequency, which are inadequate to fully represent the fault waveform. Besides, there is no
Fig. 10. Voltage distribution of single-phase fault.
Fig. 11. The instantaneous voltage acquired at bus N. 8
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Table 9 Equalization results of open-phase operation. Index
Scheme 1
Scheme 2
Scheme 3
ρ μ
0.98 1.10
0.97 1.10
0.90 27.81
and different lowest voltage location on transmission lines, even though its voltage distribution is also similar as a sine wave. The transmission lines are assumed as ideal-transposed in this manuscript, but the non-ideal transposition is almost inevitable during the transmission line construction. For such extremely long transmission lines, the non-homogeneous can induce significant inequality of the impedance relative to the ground and mutual impedance between phases, which causes the unbalance of the three phases, or even the obvious variation of power frequency wavelength. Therefore, the compensation circuit can be regarded as an invaluable method to equalize the impedance of three phases of the whole lines by adjusting the parameters of the compensation circuits. Even though the inequality is too large to be completely compensated, the compensation circuits can reduce the inequality to some extent. Compare with the other schemes of reducing the actual length of HWTL, the proposed compensation circuit presents better similarity to the transmission lines. In Ref. [16], only the air modes of the compensation circuit are considered due to its tabulated parameters of the two-port circuit. While the compensation circuit the proposed by this manuscript, which considers the compensation between phases, can achieve a better equalization result. The scheme of high frequency halfwavelength transmission lines proposed in Ref. [18] requires the ACDC-AC converters for rectification of the power frequency to a higher frequency, which is similar to the HVDC power transmission. Compared with the high frequency half-wavelength transmission lines, the tuned half-wavelength transmission lines do not need the large capacity electronic devices, which is a large economic reduction. In Table 10, the compensation circuit without capacitance between phases presented in Ref. [16], and the increasement of power frequency illustrated in [18], which decrease the actual length of the HWTL, are compared with the compensation circuit proposed in this paper.
Fig. 12. Voltage distribution of open-phase operation.
resistance in the compensation circuit to restrain the transient voltage, so the decay of oscillation induced by compensation circuit is much slower than the transmission lines, which have frequency- dependent resistance to consume the energy of high-frequency components. As the analysis at Section 2, the cut-off frequency of the compensation circuit is increasing with the rising of the number of segments, but the segment number has to reach several hundreds to achieve the cut-off frequency up to about 1 MHz, which can pass the fault induced traveling wave. Furthermore, the more segments are applied, the more distorted the transients will be. Therefore, a simple expansion of segment number just for the increase of passband is unnecessary. The compensation circuit is able to pass the power frequency component as transmission lines regardless its block or distortion of the high frequency components induced by the fault. The monitoring system based on the power frequency cannot be affected by the compensation circuit. On the other side, the fault induced transients are complete at the end of actual transmission lines, which can be acquired by the monitoring system between the compensation circuit and transmission lines. Moreover, implementation of the compensation circuit is similar to the series and shunt FACTS devices, which have never adversely influenced the operation of power system or monitoring system.
5. Conclusions The compensation circuit that considers the mutual inductance and capacitance between phases is developed from the mode system in this paper, which main findings/contributions can be summarized as follows:
4.4. Open-phase operation
(1) The analysis of frequency responses of different compensation circuits show that T and L structure compensation circuits may amplify the magnitude of some frequency components. Moreover, time delays of all kinds of compensation circuits are the same due to the identical numbers of series inductance and shunt capacitance. Based on the aforementioned characteristics, Π structure compensation circuit is more suitable for electrical length increasement than T and L structure compensation circuits. (2) In order to be similar to the natural HWTL, the capacitance between phases should be considered in the compensation circuit. Although the compensation circuit without the capacitance between phases can partly equalize the transmission lines, the compensation circuit
Assume the breakers of faulted phase have been opened and the HWTL keeps the open-phase operation to wait for the auto-reclosing, the voltage distribution and equalization results along the phase lines that are still on operation are illustrated in Fig. 12 and Table 9, respectively. The open-phase operation induces that the voltage distribution presents a curve similar as a sine wave. Clearly, the tuned HWTLs of scheme 1 and scheme 3 have almost the same voltage distribution with the natural HWTL, while they have tiny voltage difference. However, the equalization result of the tuned HWTL without the capacitance between lines is not satisfactory due to the larger voltage differences Table 10 Comparation of different methods for decreasing the actual length of HWTL. Method
Compared to the proposed compensation circuit
Compensation circuit without capacitance between phases in Ref. [16] Increasement of power frequency in [18]
The similarity to the transmission lines is inferior to the proposed compensation circuit Frequency conversion requires electronic devices with high economic costs just as HVDC
9
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including the capacitance between phases can significantly improve the equalization results by the simulation of normal operation, single-phase fault and open operation. In addition, the tuned HWTL is based on the theory and characteristics of natural HWTL, so the different voltage distribution of them may induce further technical problems. (3) The cut-off frequency of the compensation circuit can be increased by expanding the number of segments in compensation circuit, in which some high frequency components can pass through. However, a large quantity of transients, such as the traveling wave
(above 100 kHz) which needs hundreds or thousands of segments is hard to achieve. Furthermore, the expanding number of segments can induce severe signal distortion, which is unsatisfactory to increase the cut-off frequency in practice. Declaration of Competing Interest I can hereby confirm there is no interests conflict of the work to anyone.
Appendix Assume the two circuits, demonstrated in Figs. A1 and A2, are equivalent, Eq. (1) can be obtained based on Kirchhoff voltage laws. n
n
Ik1 = k= 2
Yk1(Uk
U 1) = 0
(1)
k=2
The voltage of U1 can be calculated by n
Uk =
Yk1Uk /Y
(2)
k=2
where n
Y =
Yk1
(3)
k=2
To any point i, the voltage of these two circuits can be written as
Fig. A1. Netorks with five nodes.
Fig. A2. Netorks with four nodes. 10
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Yi1(Ui
U 1) =
Y ik (Ui
Uk ) = 0
k=2 k i
(4)
Based on Eqs. (1) and (4) n k=2
Yi1(Ui
Yk1Uk / Y ) = Yi1(Ui = Yi1
n k=2 k i
n k=2
Yk1(Ui
Yk1
n k=2
Yk1Uk )/ Y
Uk )/ Y
(5)
Hence, Eq. (4) can be rewritten as n
n
Yi1
Yk1(Ui
Uk )/ Y =
k=2 k i
Yk1 (Ui
Uk )
k=2
(6)
Due to the equivalence of Eq. (6) at any voltage, the admittance of these two circuits can be written by n
Y ij = Yi1Yj1/Y
= Yi1Yj1/
Yk1
(7)
k=2
Assume Y23 = Y24 = Y25 = Y0_com , Y34 = Y45 = Y35 = Ycom , Eq. (8) can be obtained from Eq. (7) as
Y12 =
2 Y0_com + 3Y0_com Ycom
(8)
Ycom
Y13 = Y14 = Y15 = Y0_com + 3Ycom If Y0_com = sC0_com and Ycom = sCcom, Eq. (23) can be obtained by
CN_com =
2 C0_com
Ccom
+ 3C0_com
CP_com = C0_com + 3Ccom
(10)
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