1. Introduction
The HVDC transmission mode based on a modular multilevel converter has many advantages, such as the flexible consumption of large-scale renewable energy, power supply to a passive network, independent control of the active/reactive power of the system, and the absence of the need to change the voltage polarity when the system power flow is reversed [
1,
2,
3]. It has significant advantages in offshore wind farms, grid-connected new energy power generation, isolated island and weak grid power supply, asynchronous grid interconnection, and other projects and has broad development prospects [
4,
5].
In recent years, a number of long-distance flexible DC transmission projects have been completed and put into operation [
6,
7,
8]. On 8 August 2023, China’s Hayu ±800 kV DC project officially began construction, with a total length of 2290 km of DC lines. In November, the Ganzhe DC project site conducted a selection survey on the use of fully flexible DC transmission technology, with a DC line length of 2368 km.
The continuous development and increasing application of MMC-HVDC have greatly improved the capacity, transmission distance of the system, and also significantly increased the probability of DC system short-circuit faults. In addition, the characteristics of MMCs, such as their small inertia and many energy storage components, lead to high rates of increase in short-circuit current and large overcurrent amplitudes, making it difficult for DC circuit breakers to be turned off. The increase in transmission distance makes an equivalent model of accurate and simple transmission lines particularly important for short-circuit current calculations. The above problems limit the further promotion and development of MMC-HVDC power grids [
9,
10,
11,
12]. Thus, the study of an effective equivalent model of long-distance transmission lines and the quantitative calculation of the MMC-HVDC short-circuit current on the DC side considering the line distribution parameters can not only provide a theoretical basis for the planning and design of multi-terminal flexible HVDC systems and the optimization of control and protection strategies, but it can also have important theoretical and engineering application value for seeking ways to reduce the fault current level.
There have been many studies on the calculation of the short-circuit current on the DC side of MMC-HVDC [
13,
14,
15]. Based on a ±500 kV flexible DC transmission engineering model with symmetrical bipolar MMCs, authors [
16] studied the circuit topology and its transient characteristics when different kind of faults occur. Then, they compared the effect of AC and DC circuit breakers to clear and recover various faults. Based on a single MMC short-circuit equivalent circuit, authors [
17] studied short-circuit current calculation methods for a single converter valve considering a fault current limiter and arrester and analyzed the fault characteristics under different working conditions. Based on the RLC equivalent model, the pre-fault matrix of the multi-terminal converter station was derived from a single converter station [
18], and the corrected fault matrix was established with several fault locations in order to solve the DC current of each branch. However, the method is not universal for the treatment of various faults in the MTDC power grid. In reference [
19], considering that the MMC adopts constant DC voltage control, a short-circuit current calculation method with a differential equation was proposed. In reference [
20], the dynamic method of a power supply and DC voltage controller based on a current differential equation was improved to provide the possibility of analyzing fault currents in the power grid under different operating conditions (such as fault resistance and limit inductance value). In reference [
21], a fault current calculation method based on a coupled differential equation was proposed, and the short-circuit current of the MMC with and without locking was calculated using a complex differential equation. In reference [
22], the authors provide a calculation method for the short-circuit current before MMC blocking by establishing a linearized model of a modular multilevel converter, but different model forms need to be built for different faults.
HVDC transmission systems usually use overhead lines for transmission. Due to the frequency change characteristics of line parameters, when the transient electric gas is transmitted along the transmission line at different frequencies, its propagation speed and attenuation coefficient are significantly different [
23,
24]. The authors of reference [
25] greatly increased the difficulty of the analytical calculation for flexible transmission system DC short-circuit currents. In reference [
26], a transmission line model in the phase domain was established based on the relationship between phase current and voltage at the transmitting and receiving ends of single-phase lines, and the relationship established by the ABCD matrix was extended to polyphase lines. In reference [
27], the distribution parameter characteristics and frequency variation characteristics of transmission lines were considered, and the AC system with long-distance transmission cables was made equivalent by using the vector matching method. However, the distributed parameter line models provided by these studies mainly focus on AC transmission systems.
For HVDC systems, reference [
28] considers the influence of mutual inductance, uses the measured inverse sequence current and fault inverse sequence current to calculate the phase and amplitude of the voltage, and performs fault location by solving the phase mutation point of the fault location function. The authors of reference [
29] sharply consider the effects of distribution parameters and frequency-related line parameters, propose an improved frequency-dependent line model based on convolution, and design a convolution-based time-domain fault location algorithm. However, the above literature mainly considers the fault location problem in the equivalent of the distributed parameter line model, and it does not study the transmission line model suitable for the calculation of DC short-circuit current.
Considering the distributed parameter model, a method for calculating the short-circuit current of a flexible HVDC system with distributed parameters is proposed. Firstly, the distributed parameter circuit model of a line unit length element is established, the transmission line equation is derived, and the steady state expression of voltage and current is obtained using a wave equation, to calculate the line impedance. Different line impedance equivalent models are studied, and different model equivalents for different fault distances are proposed to reduce the calculation error. The correctness of the calculation method is verified by comparing the simulation value with the calculated value.
2. Transmission Line Equation
The unit length parameters of the transmission line include the resistance, reactance, conductance, and susceptance of the unit length line. In the calculation, the split wire may be replaced with an equivalent wire. The reactance value and susceptance value can be determined according to the expression of the single-core wire. The resistance and conductance values are independent of the shape of the transmission line, where the resistance value is determined by the number of wires per pole, while the conductance value depends on the corona active power loss and leakage on the insulation, and due to their small values, they are usually ignored in practical calculations.
The theory of a uniform transmission line assumes that the parameters of any section of the power are evenly distributed along the transmission direction. It has nothing to do with the line structure and the transverse and longitudinal distribution of the electromagnetic field, and it only considers the changes in current and voltage along the line. Taking unipolar lines as an example, the distributed parameter equivalent model of long-distance transmission lines is studied. A power line with a length of x is divided into many infinitesimal length elements dx; then, each length element of dx has resistance
Rldx and inductance
Lldx and has capacitance
Cldx and conductance
Gldx between the line and the ground. The structure is shown in
Figure 1.
Let the unit length resistance, inductance, conductance, and capacitance be constant, and the transmission line equation applicable to any excitation source can be listed as shown in Equation (1).
In a distributed parameter line, the voltage to ground and the line current at any location are functions of time and distance, in line with the principle of electromagnetic wave propagation along the route; then, the wave process of a unipolar line can be expressed through partial differential equations, as shown in Equations (2) and (3):
The wave equation can be obtained from Equations (2) and (3):
where
v is the propagation speed along the wave, and
.
The D’Alembert solution of Equations (4) and (5) is as follows:
where
u+ and
i+ are the voltage and current with a velocity
v propagating in the positive direction of
x;
u− and
i− are voltages and currents with a velocity of
v propagating in the opposite direction of
x. The individual
u+,
u−,
i+, and
i− are all solutions to the wave equation, and they propagate independently, complementally interfering.
The characteristic equation in the time domain can be obtained from Equations (6) and (7):
When the system is in a stable state, the voltage and current can be expressed as follows:
where
is the propagation coefficient,
is the characteristic impedance,
is the line impedance per unit length, and
is the admittance per unit length.
A simple two-terminal MMC-HVDC is shown in
Figure 2. The current is transmitted from MMC 1 to MMC 2, and the total length of the transmission line is
L. When the voltage and current at one end are known, the relationship between the voltage and current at both ends of the line can be expressed, as shown in Formulas (11) and (12).
where subscript A represents the amount of electricity propagating in the positive direction, and subscript B represents the amount in the opposite direction.
The above formula is the distribution parameter expression of the line, but it does not consider the frequency change. Since the resistance per unit length of the transmission line is positively correlated with the frequency, and the inductance is negatively correlated with the frequency, that is, different frequencies make the impedance of the line different, showing different transmission characteristics, it is necessary to correct the line impedance.
where
is the resistance per unit length of the line when the frequency is f
0.
By converting the voltage and current relationship between the two ends of the line into the time domain, the equivalent circuit of the line distribution parameter model can be obtained, as shown in
Figure 3. The first and last ends of the line are connected in parallel by an initial current source to the ground branch and the equivalent impedance, and the first and last ends are physically independent of each other and are interconnected through the electromagnetic relationship of the current source.
The above model does not consider the line frequency characteristics, and a lossless line model is established according to the traveling wave method, where the equivalent impedance and current source can be expressed as follows:
In the formula, L0 and C0 are the fundamental frequency inductance and capacitance of the per length element line, respectively.
5. Simulation Verification
A two-terminal flexible DC transmission system model is built based on the PSCAD/EMTDC, as shown in
Figure 8. The line length
L is 2000 km, and the simulation step is 50 μs. The overhead line frequency-dependent phase domain model is adopted for the DC transmission line.
Table 2 and
Table 3 list the parameters of the MMC station and transmission line. A gold pole-to-pole short-circuit fault occurred on the line between MMC 1 and MMC 2.
When the distance between MMC 1 and the fault is
L = 0 and there is no transmission line, the short-circuit current can be calculated directly using an analytical expression. When the distance between the fault point and the converter station is not 0, the short-circuit current should be calculated according to the flow diagram in
Figure 7, considering the distribution parameter characteristics and frequency variation characteristics of the line.
Figure 9a–e show the comparison of simulation values with those calculated with the lumped parameter model and the distributed parameter model under the working conditions of 0 km, 600 km, 1000 km, 1600 km, and 2000 km between the fault point and MMC 1, respectively.
Table 4 shows the errors between the simulated values for different fault distances and the calculated values for the line lumped and distributed parameter models, which is the average of the error at each time within 10 ms.
From the above comparison results, when short-circuit faults occur at different locations, the distributed parameter model considering frequency variation characteristics can calculate the short-circuit current, and the error between the calculated current and the simulated current is small. With an increase in the distance between the MMC and the fault, the increase in short-circuit current decreases gradually. When the fault occurs at the outlet of the converter station, the analytical calculation values of the short-circuit current determined using the lumped parameter model and distributed parameter model can be accurately fitted to the simulation values. But when the fault is far away from the converter station, the analytical calculation of the short-circuit current using the lumped parameter model has a greater average error within 10 ms. When the line length is set to 0 km by the distributed parameter model, there is a small ripple in the simulation values of the short-circuit current, which makes the calculation error of the distributed parameter seem larger than that of the lumped parameter. But it can be ignored.
To further verify the effectiveness and universality of the above methods, a six-terminal MMC-HVDC transmission system is built on the PSCAD. The topological structure is demonstrated in
Figure 10, and the parameters of the MMC and transmission line are shown in
Table 5 and
Table 6, respectively. When the bipolar short-circuit fault occurs on cable 12, the analytical calculation method proposed in this paper is used to obtain the current of the line. The comparison between the calculated current and the simulation one is shown in
Figure 11. From this, we could learn that this method is also suitable for multi-terminal flexible HVDC systems with complex structures, and the maximum error between the short-circuit current calculated by the distributed parameter method and the simulation value within 10 ms is significantly smaller than that calculated by the lumped parameter. The correctness of the proposed calculation method is verified.
6. Conclusions and Limitations
The DC short-circuit analytical expression of a flexible HVDC system considering the distributed parameter model is an important component that must be configured in the development of DC power networks. In order to obtain an appropriate method for calculating the short-circuit current of long-distance transmission lines, the transmission line equation for transmission lines is first studied, the transmission line model of unit length is established, the relationship between voltage and current at both ends and line parameters is deduced, and then the line model is made approximately equivalent. A method for calculating the MMC-HVDC short-circuit current on the DC side considering the distributed parameter model for long-distance transmission lines is presented. The main conclusions are as follows:
With an increase in the transmission line length, the error between the equivalent impedance calculated with the lumped parameter model and the equivalent impedance calculated with the distributed parameter model increases significantly. When the line length is less than 300 km, the equivalent reactance calculated with the lumped parameter model and the distributed parameter model is basically equal; when the line length exceeds 350 km, the equivalent reactance calculated with the lumped parameter model and the distributed parameter model is basically equal. The deviation in the equivalent reactance of the two models increases gradually with an increase in the line and reaches about twice the error at 1000 km.
When the line length is less than 300 km, the lumped parameter model can be used for direct calculation; when the line length is 300–500 km, the circuit impedance is calculated with the modified coefficient method to solve the short-circuit current. When the line length is greater than 500 km, the Gorev method line equivalent model is used for calculation.
Upon comparing the simulation and analytical calculation values for different fault distances, the results show that the analytical calculation values can better characterize the fault current characteristics and effectively reduce the error of the lumped parameter model.
Although this research found some interesting and meaningful conclusions, there are still some limitations to this study. For instance, we only studied the case of short-circuit faults in DC lines, not lightning discharge, open-circuit faults, and so on. In the future, further research can be conducted to study the effects of disturbances.