1 Introduction

Half-wavelength alternating current transmission (HWACT) refers to a kind of AC transmission technology whose transmission distance is equal to a half of the wavelength under the fundamental frequency [1]. With the concept of global energy interconnection being proposed, HWACT technology becomes of more interest. In 2015, the State Grid Corporation of China (SGCC) launched a series of research projects on HWACT technology in order to seek alternatives for long-distance power transmission besides high voltage direct current (HVDC) transmission.

The equivalent reactance of a HWACT line is theoretically equal to zero. This can largely reduce the electrical distance between both ends of the line [2, 3]. Because of this, HWACT is suitable for long-distance and high-capacity transmission [4,5,6,7,8]. For the traditional line, the resistance is far less than the reactance. Thus, the active power is strongly correlated with the phase angle of the node voltage whereas the reactive power is strongly related to the amplitude of the node voltage [9,10,11,12,13,14]. As the length of the line reaches the half-wavelength, the power-voltage characteristic becomes different. Under the assumption that the line is lossless, the voltage amplitudes of both ends are equal and the phase angle difference is 180 degrees for the HWACT line [1,2,3, 15]. This indicates that the terminal voltages are not related to the transmission power. However, further investigation is required to verify the correctness of this conclusion. The power-voltage characteristics also determine the overvoltage on the HWACT line. These could be serious when the transmission power is high [3, 16]. If the power factor is equal to 1, the overvoltage at the midpoint of the line is roughly equal to the ratio of the active power to the natural power [1,2,3, 17, 18].

HWACT technology can also be used to enhance synchronous power grids. Reference [19] proposes the concept of building stereoscopic power grids using multiple HWACT lines. The HWACT lines share a common sending end and connect to different terminals in one power grid. The single point-double terminals (SPDT) system has been built in [19] and it is supposed to be the most typical system reflecting the networking capability of HWACT. The simulations demonstrate that the stability of the power grid can be enhanced effectively because the HWACT lines shorten the electrical distances between the terminals, changing the grids into a stereoscopic structure from a plane structure. Since it is a relatively new field, there are no more relevant references except [19]. Thus, some new features of the SPDT system are not yet completely clear. The power flow distribution on the HWACT lines is one of the key issues which need further study. The situation of the power flow distribution is quite complex in the SPDT system compared with that of a single line. Thus, it is necessary to study the power flow distribution characteristics of the SPDT system in depth.

Aiming at the above-mentioned problems, this paper derives and validates the power-voltage characteristics as well as the power distribution of the HWACT system. According to the equations of the distributed parameter model, the power-voltage characteristics of a single HWACT line are derived. The “reverse characteristics” of the HWACT line are found, opposite to those of regular short transmission lines. Then, the concept and definition of the penetrating power (PP) in a SPDT system is proposed. The corresponding relationship among PP, the terminal bus voltages and the supplied power can be obtained based on the principles and characteristics of PP. The research indicates that the relationship between PP and terminal voltages also shows “reverse characteristics”. The simulations are carried out to validate the theoretical analysis under different conditions. The results indicate that demanding requirements should be made for the precision control of a HWACT system.

2 Power-voltage characteristics of a single HWACT line

A single AC transmission line is shown in Fig. 1, where \(\dot{U}_{2}\) and \(\dot {U}_{1}\) are the voltage phasors of both terminals, \(\dot {I}_{2}\) and \(\dot {I}_{1}\) are the current phasors of both terminals, P and Q are the active and reactive power from the terminal 2 to 1, respectively.

Fig. 1
figure 1

Diagram of a single AC transmission line

Full size image

The formula of the distributed parameter model can be written as [2]:

$$\left[ \begin{aligned} \dot {U}_{1} \\\dot {I}_{1} \\ \end{aligned} \right] = \begin{bmatrix}\cosh (\gamma l) \, & Z_{s} \sinh (\gamma l) \\\frac{1}{{Z_{s} }}\sinh (\gamma l) & \cosh (\gamma l) \end{bmatrix} \left[ \begin{aligned} \dot {U}_{2} \\ - \dot {I}_{2} \\ \end{aligned} \right]$$
(1)

where γ is the transmission coefficient; Zs is the wave impedance; l denotes the length of the line. γ can be represented as α+jβ = \(\sqrt {z_{k} y_{k} }\), Zs can be represented as \(\sqrt {z_{k} /y_{k} }\). zk and yk represent the impedance, admittance per unit length, respectively. zk, yk can be written as rk + jxk, gk + jbk, respectively. It can be found that rk ≪ xk and gk ≪ bk.

Assuming that \(\dot {U}_{2} = U_{2} \angle 0^{ \circ }\) and \(\dot {I}_{2} = I_{2} \angle - \varphi\), it can be found that:

$$\dot {U}_{1} = \cosh (\gamma l)U_{2} - Z_{s} \sinh (\gamma l)\dot {I}_{2}$$
(2)
$$\left\{ \begin{aligned} P = U_{2} I_{2} \cos (\varphi ) \hfill \\ Q = U_{2} I_{2} \sin (\varphi ) \hfill \\ \end{aligned} \right.$$
(3)
$$\dot {I}_{2} = \frac{P - \text{j}Q}{{U}_{2}}$$
(4)

According to (2)-(4), the equation of power-voltage characteristic can be represented as follows:

$$\dot {U}_{1} = \cosh (\gamma l)U_{2} - Z_{s} \sinh (\gamma l)\frac{P - \text{j}Q}{{U}_{2}}$$
(5)

Then, the coefficient cosh(γl) and sinh(γl) can be expanded as:

$$\left\{ \begin{aligned} \cosh (\gamma l) = \frac{1}{2}(\text{e}^{\alpha l} + \text{e}^{ - \alpha l} )\cos (\beta l) + \text{j}\frac{1}{2}(\text{e}^{\alpha l} - \text{e}^{ - \alpha l} )\sin (\beta l) \hfill \\ \sinh (\gamma l) = \frac{1}{2}(\text{e}^{\alpha l} - \text{e}^{ - \alpha l} )\cos (\beta l) + \text{j}\frac{1}{2}(\text{e}^{\alpha l} + \text{e}^{ - \alpha l} )\sin (\beta l) \hfill \\ \end{aligned} \right.$$
(6)

In order to analyze the effects of α and β, the derivation of eαlcos(βl) and eαlsin(βl) is shown as follows:

$$\left\{ \begin{aligned} \frac{{{\text{de}}^{\alpha l} \cos (\beta l)}}{{\text{d}l}} = \text{e}^{\alpha l} (\alpha \cos (\beta l) - \beta \sin (\beta l)) \hfill \\ \frac{{{\text{de}}^{\alpha l} \sin (\beta l)}}{{\text{d}l}} = \text{e}^{\alpha l} (\alpha \sin (\beta l) + \beta \cos (\beta l)) \hfill \\ \end{aligned} \right.$$
(7)

According to (7), β makes more contribution to parameter-growth if α/β < |tan(βl)| < β/α. Since β ≫ α, the effect of α can be ignored, which is consistent with the hypothesis for the lossless line. On the other hand, if |tan(βl)| < α/β or |tan(βl)| > β/α, the effect of α should not be ignored. As for the HWACT line, the value of βl is equal to π, so (5) can be simplified to:

$$\dot {U}_{1} = - \frac{1}{2}(\text{e}^{\alpha l} + \text{e}^{ - \alpha l} )U_{2} + \frac{1}{2}(\text{e}^{\alpha l} - \text{e}^{ - \alpha l} )Z_{s} \frac{{P - \text{j}Q}}{{U}_{2}}$$
(8)

Since the resistance is much smaller than the reactance per unit length, the imaginary part of the wave impedance Zs is very small. Thus, if the value of U2 is determined, the real part of \(\dot {U}_{1}\) is related to the active power whereas the imaginary part is related to the reactive power.

Taking the 1000 kV ultra-high voltage (UHV) typical circuit as an example (8×500 mm2 steel-cored aluminum wire), the calculated parameters are: zk = (8.01 × 10−6 + j2.631 × 10−4)Ω/m, yk = j4.3448 × 10−9 S/m, α = 1.6273 × 10−8 m−1, β = 1.0693 × 10−6 m−1. So the exact half-wavelength length can be calculated as 2938.3 km. The voltage base UB is 1050 kV and the power base SB is 100 MVA. So (8) can be modified as:

$$\dot {U}_{1} = - 1.00114U_{2} + 1.068 \times \text{10}^{ - 3} \frac{P}{{U_{2} }} - \text{j}1.068 \times \text{10}^{ - 3} \frac{Q}{{U_{2} }}$$
(9)

Since the coefficients of P, Q are small, the real part of (9) is much larger than the imaginary part when P, Q are in a normal range. Thus, the amplitude of \(\dot {U}_{1}\) has a higher correlation with the active power whereas the phase angle is more related to the reactive power. This feature shows “reverse characteristics” which is opposite to those of regular short transmission lines. Based on the previous derivation, it can be inferred that Zssinh(γl) is the equivalent series impedance of π circuit of the line. If βl = π, the equivalent reactance of Zssinh(γl) is equal to zero in theory, so the coefficients of P and Q only reflect the effect of the equivalent resistance.

According to (9), the amplitude and phase angle variation of \(\dot {U}_{1}\) is shown in Figs. 2 and 3. The value of U2 is set at 1.0 p.u..

Fig. 2
figure 2

Curves of terminal voltage amplitude varied with power for HWACT line

Full size image
Fig. 3
figure 3

Curves of terminal voltage phase angle varied with power for HWACT line

Full size image

As shown in Figs. 2 and 3, if P remains unchanged and Q ranges from − 50 to 50 p.u. (− 5000-5000 Mvar), the value of U1 ranges from 1.0012 to 1.0026 p.u., and the maximum variation of the voltage amplitude is about 0.0014 p.u. In this case, the value of angle changes from 177° to 183°, and the maximum variation of the phase angle is about 6°. According to the same figures, if Q remains unchanged and P changes in the same interval, the maximum variation of the voltage amplitude is approximately 0.1 p.u. (from 0.95 to 1.05 p.u.), while the phase angle remains 180°. The results verify the previous discussion to some extent. However, it also indicates that the inference from the hypothesis of the lossless line (the voltage amplitude of each terminal is equal and the phase angle difference is 180°) is not strictly correct. As for the HWACT line, the resistance cannot be ignored in some circumstances because it has impact on “reverse characteristics” opposite to those of regular short transmission lines.

3 Power-voltage characteristics of SPDT system

To derive the power-voltage characteristics of a SPDT system, two HWACT lines are adopted here to constitute the system, as shown in Fig. 4. Point c connects to points 1, 2 in a synchronous power grid by two HWACT lines, respectively.

Fig. 4
figure 4

Diagram of SPDT system

Full size image

Sc represents the supplied output power at point c. Sc1 and Sc2 represent the transmission power from point c to 1, 2, respectively. In this scenario, the electrical distance between the two separated points (1, 2) becomes shorter by using the HWACT line, so the mutual supporting capability can be improved. According to the analysis above, the PP which flows through the HWACT lines will be generated between point 1 and point 2 when there exists a voltage difference. The PP combines with Sc, leading to uneven distribution of the actual transmission power on each line. In this paper, PP is defined as:

$$\Delta S_{pe} \triangleq \frac{1}{2}(S_{c1} - S_{c2} )$$
(10)

It can be obtained as:

$$\left\{ \begin{aligned} S_{c1} = \frac{1}{2}S_{c} + \Delta S_{pe} \hfill \\ S_{c2} = \frac{1}{2}S_{c} - \Delta S_{pe} \hfill \\ \end{aligned} \right.$$
(11)

According to (1), the relationship between voltage and current can be written as:

$$\left[ \begin{aligned} \dot{I}_{c1} \\ \dot{I}_{1} \\ \end{aligned} \right] = \begin{bmatrix}B^{ - 1} A & - B^{ - 1} \\C - AB^{ - 1} A & AB^{ - 1} \end{bmatrix}\left[ \begin{aligned} \dot {U}_{c} \\ \dot {U}_{1} \\ \end{aligned} \right]$$
(12)

where \(\dot {I}_{c1}\) is the current flowing from point c to terminal 1; A, B, C denote cosh(γl), Zssinh(γl), and sinh(γl)/Zs, respectively.

The transmission power on Line 1 is:

$$S_{c1} = \dot {U}_{c} \dot {I}_{c1}^{*} $$
(13)

where the superscript * denotes the conjugate of the phasor. Combined with (12), (13) can be expanded as:

$$\begin{aligned} S_{c1} &= \dot {U}_{c} (B^{ - 1} A\dot {U}_{c} - B^{ - 1} \dot {U}_{1} )^{*} \hfill \\ &= (B^{ - 1} A)^{*} U_{c}^{2} - (B^{ - 1} )^{*} \dot {U}_{c} \dot {U}_{1}^{*} \hfill \\ \end{aligned}$$
(14)

Similarly, the transmission power on Line 2 is obtained as:

$$S_{c2} = (B^{ - 1} A)^{*} U_{c}^{2} - (B^{ - 1} )^{*} \dot {U}_{c} \dot {U}_{2}^{*} $$
(15)

Thus, the PP can be derived as:

$$\Delta S_{pe} = \frac{1}{2}(B^{ - 1} )^{*} \dot {U}_{c} (\dot {U}_{2}^{*} - \dot {U}_{1}^{*} )$$
(16)

The supplied output power is written as:

$$\begin{aligned} S_{c} &= S_{c1} + S_{c2} \hfill \\ &= 2(B^{ - 1} A)^{*} U_{c}^{2} - (B^{ - 1} )^{*} \dot {U}_{c} (\dot {U}_{2}^{*} + \dot {U}_{1}^{*} ) \hfill \\ \end{aligned}$$
(17)

As for the HWACT line, the value of βl is equal to π, so the coefficients A, B can be simplified as real numbers:

$$\left\{ \begin{aligned} &A = \cosh (\gamma l) = - \frac{1}{2}(\text{e}^{\alpha l} + \text{e}^{ - \alpha l} ) = A^{*} \hfill \\ &B = Z_{s} \sinh (\gamma l) = \frac{1}{2}Z_{s} (\text{e}^{ - \alpha l} - \text{e}^{\alpha l} ) = B^{*} \hfill \\ \end{aligned} \right.$$
(18)

Combining (16)-(18), \(\dot {U}_{\text{c}} \) is eliminated and the relationship among Sc, \(\dot {U}_{1}\), \(\dot {U}_{2} \) and ∆Spe can be obtained as:

$$S_{c} = 8AB\left| {\frac{{\Delta S_{pe} }}{{\dot {U}_{2} - \dot {U}_{1} }}} \right|^{2} - 2\Delta S_{pe} \left( {\frac{{\dot {U_{2} } + \dot {U}_{1} }}{{\dot {U}_{2} - \dot {U}_{1} }}} \right)^{*}$$
(19)

1) Assuming that the terminal voltages are known. According to (19), ∆Spe varies with Sc nonlinearly. For example, the conditions are set as follows: \(\dot {U_{1} }\) = 0.95∠− 3°, \(\dot {U_{2} }\) = 1.0, the power factor is 1 (reactive component Qc = 0). Ppe and Qpe represent the active and reactive components of PP, respectively. The curves of Ppe, Qpe varied with Pc are presented in Fig. 5, which is produced according to (19). It can be seen that the PP varies with the supplied power non-linearly over a large range (− 500-500 p.u.) whereas the rate of change is small. Thus, the amplitude change of the PP is small when the supplied power ranges from − 100 to 100 p.u., as is shown in the red frame.

Fig. 5
figure 5

Curves of PP varied with source power

Full size image

2) Assuming that the supplied power is known. In the typical case of Sc = 0, there is only PP flowing through the line. Thus, the equation can be obtained based on (18) and (19):

$$S_{pc} = \frac{1}{{Z_{s}^{*} (\text{e}^{2\alpha l} - \text{e}^{ - 2\alpha l} )}}(U_{2}^{2} - U_{1}^{2} + \text{j}2U_{1} U_{2} \sin (\varphi_{12} ))$$
(20)

where φ12 denotes the phase difference between terminals 1 and 2.

The active and reactive components of PP can be written as:

$$\left\{ \begin{aligned} P_{pe} = \frac{{U_{2}^{2} - U_{1}^{2} }}{{\left| {Z_{s} } \right|(\text{e}^{2\alpha l} - \text{e}^{ - 2\alpha l} )}} \hfill \\ Q_{pe} = \frac{{2U_{1} U_{2} \sin (\varphi_{12} )}}{{\left| {Z_{s} } \right|(\text{e}^{2\alpha l} - \text{e}^{ - 2\alpha l} )}} \hfill \\ \end{aligned} \right.$$
(21)

This shows that the active component of PP is only linearly related to the square error of voltage amplitude between the terminal points if there is no power being sent. On the other hand, the reactive component of PP is determined by both the voltage amplitude and the phase difference.

If U2 > U1, then Ppe > 0; that is, the active PP flows from the higher-voltage bus to the lower-voltage bus. On the other hand, if φ12 > 0, then Qpe > 0. Therefore, the reactive PP flows from the smaller-phase angle bus to the larger-phase angle bus. Furthermore, the phasor diagrams of U1, U2 and Uc can be obtained, which are presented in Fig. 6.

Fig. 6
figure 6

Diagram of relationship among node voltage vectors in SPTD system

Full size image

The above analysis indicates that the PP between the terminal points can be controlled by adjusting the voltage amplitude and phase angle of each terminal. In addition, the active and reactive power can be decoupled. Ideally, the value of PP should be controlled at zero to guarantee the maximum transmission power. The active PP can be controlled by arranging the amount of reactive power compensation at the terminal point properly. This contributes to adjusting the voltage amplitude. As for the reactive PP, the phase angle can be controlled by adjusting the unit operation mode at the adjacent areas of terminal points.

4 Simulation verification

In order to verify the theoretical analysis of the characteristics of a single HWACT line and SPDT system, PSD-BPA [20] is used to simulate the load flow distribution in a power grid. The HWACT line is modeled by 20 π sections connecting with each other [18]. The line length is set as 2938.3 km.

4.1 Simulation of single HWACT line

As shown in Fig. 7, the simulation is carried out based on the case of planning UHV power grids of the SGCC for the year 2020. The HWACT line with voltage level of 1050 kV connecting the generator and a terminal bus is added to the original data. The location of the terminal bus is selected as Jingmen. Relevant parameters are presented in Fig. 7, where Sg and Ug represent the rated capacity and voltage of the sending unit, respectively, St and Xk are the rated capacity and leakage reactance of the transformer, respectively, Sb = 100 MVA and Ub = 1050 kV are the base capacity and base voltage of the system, respectively, Zeq and Yeq refer to the series impedance and the parallel admittance of each equivalent π section, respectively.

Fig. 7
figure 7

Simulation case of single HWACT line connected to power grid

Full size image

The bus of Jingmen station is set as the balance node, whose voltage amplitude U2 is set as 1.0 p.u., and the phase angle is 0°. To verify the “reverse characteristics” of the HWACT line, the theoretical and simulation results of the head-end voltage U1 are compared under various P and Q, which are shown in Table 1.

Table 1 Comparison of theory and simulation results for single HWACT line
Full size table

According to Table 1, if the reactive power remains unchanged and the active power is adjusted step-by-step, the terminal voltage mainly displays amplitude variation. However, if the reactive power is adjusted with the invariability of active power, the terminal voltage is changed mainly in phase angle. The amplitude errors between the theoretical and simulation results are in the 10-3 order of magnitude, and the phase angle errors are in the 10-2 order of magnitude. It can be seen that the theoretical and simulation results have high similarity. Thus, the correctness of the theory can be verified.

4.2 SPTD system

Based on the case of a single HWACT line, the second HWACT line is connected to the system. The location of the terminal is Nanyang, as shown in Fig. 8. In addition, the parameters of the second HWACT line are the same as the first line in the SPTD system. The relevant parameters are shown as: Zs = 246.107 Ω, α = 1.6273 × 10−8 m−1. C refers to the location of the supplied output power.

Fig. 8
figure 8

Simulation solution for SPTD system

Full size image

Considering a simple situation where there is no power source, the value of PP can be calculated based on (20). The comparison of the theoretical and simulation results for PP is presented in Table 2. U2 and U1 are the voltage of Jingmen, Nanyang station, respectively. φ12 represents the phase angle difference. Pcr and Qcr represent the active and reactive PP, respectively.

Table 2 Comparison of theory and simulation results for SPTD system
Full size table

As can be seen in Table 2, the PP will change over a wide range if the value of the terminal voltages is adjusted. The simulation results have shown satisfactory agreement with theory. The maximum errors of the active and reactive PP are 0.92 p.u. (92 MW), 0.21 p.u. (21 Mvar), respectively.

The relation between the active PP and terminal voltages is presented in Fig. 9. It can be inferred that the active PP varies linearly with \(U_{2}^{2}\)–\(U_{1}^{2}\). This is identical to the theoretical results. The relation between the reactive PP and φ12 is shown in Fig. 10. The phase angle ranges from −3.8° to −0.8°. Thus the simulation and theory results have high similarity. Since the value of φ12 is small, sin(φ12) ≈ φ12. Thus, the reactive PP is strongly correlated to the phase angle difference between the terminal buses.

Fig. 9
figure 9

Active PP varied with amplitude of terminal voltages

Full size image
Fig. 10
figure 10

Reactive PP varied with phase angle of terminal voltages

Full size image

According to Figs. 9 and 10, the PP shows great variation with the voltage amplitude and the phase angle at terminals. That is, a little voltage difference can lead to a large PP. It can be seen that the active power is more correlated with the amplitude of the bus voltage whereas the reactive power is more related to the phase angle.

Since the supplied power may not be zero in reality, simulations are also performed with different supplied power, as shown in Fig. 11. It can be seen that PP is mainly related to the terminal voltages, and the variation ratio with Sc is very small. In Fig. 11, with the variation scale 100 p.u. of Sc, the variation scale of PP is less than 3 p.u.. The conclusion is consistent with the theoretical curve of Fig. 5.

Fig. 11
figure 11

PP varied with terminal voltages in different supplied powers

Full size image

To further verify the proposed theory, more simulations are carried out under different conditions, as shown in Figs. 12 and 13. The characteristics of PP are validated with different voltage grades and propagation coefficients. Thus, the accuracy and adaptiveness of the theoretical analysis can be proved.

Fig. 12
figure 12

PP varied with terminal voltages in different voltage grades

Full size image
Fig. 13
figure 13

PP varied with terminal voltages in different propagation coefficients

Full size image

5 Conclusion

In this paper, the “reverse characteristics” of the HWACT line and the concept of PP have been proposed.

1) The HWACT line shows “reverse characteristics” opposite to those of regular short transmission lines. The active power is more correlated with the amplitude of the bus voltage whereas the reactive power is more related to the phase angle. From this perspective, HWACT is similar to DC transmission. Thus, the resistance should not be ignored in the calculation and analysis of the HWACT line.

2) The power flow distribution of the SPTD system is analyzed. This helps reveal the characteristics of the PP on the HWACT line. PP is much more related to terminal voltages than the supplied power. The active PP is linearly related to the square error of amplitudes of the terminal voltages whereas the reactive PP is highly determined by the phase angle difference of the terminal voltages. Little difference of the voltage amplitude and phase angle will lead to a large PP. Thus we can infer that different control methods should be adopted in the HWACT system. In addition, the accuracy and reliability of the control is demanding, and the terminal locations of the HWACT lines need to be selected carefully.

3) This paper provides a preliminary study on the related problems of the HWACT system. This should be helpful for the application of HWACT technology. In future work, control methods for the HWACT system as well as the characteristics of the mixed HWACT system (such as the grid-to-grid system in which two independent power grids are connected by several HWACT lines) will be further researched.