1. Introduction
High-voltage direct current (HVDC) transmission system offers significant advantages for efficient and reliable large-scale power transmission in interconnecting power systems. HVDC transmission is suitable for long-distance, large-capacity power transmission and can be applied to interconnecting asynchronous systems, submarine cable transmission, and stable AC systems [
1,
2,
3]. HVDC has been proven to be a preferable solution for addressing the renewable energy integration issue [
4].
In HVDC systems, two primary converter technologies are employed: line-commutated converters (LCCs) and voltage-source converters (VSCs). The LCC technology offers advantages such as low losses for long-distance transmission. However, it faces challenges such as commutation failures, harmonics generation, and the need for reactive power compensation and polarity reversal for power flow reversal [
5,
6,
7,
8]. On the other hand, VSC technology provides benefits like independent active and reactive power control, improved dynamics, and stable operation without large harmonic filters. However, it incurs higher construction costs, lower power ratings, and typically higher operational losses [
9,
10,
11]. Combining the strengths of LCC and VSC, the hybrid HVDC system presents a promising solution for power transmission with reduced losses and costs, as well as enhanced control flexibility [
12,
13]. Compared with conventional VSC, a modular multilevel converter (MMC) offers lower voltage harmonics and lower normal operating losses [
14]. The hybrid HVDC technology has gained attention in recent years due to several ongoing projects in China, such as the Baihetan–Jiangsu system [
15] and the Wudongde–Yunnan system [
16].
The study of AC/DC power flow calculation dates back to the 1950s. Since then, extensive research has been conducted to address AC/DC power flow issues in power systems using LCC [
17,
18,
19]. This research has served as a pioneer foundation for subsequent studies on power flow calculations involving various types of converters, such as VSC and current-source converters (CSCs). The unified power flow calculation method was first reported in [
20], which described an application for a two-terminal VSC system. Reference [
21] introduces a generalized steady-state mathematical model for multi-terminal VSC systems with arbitrary topologies.
The power flow calculation methods for AC/DC systems can be classified into two categories: the unified and the sequential methods. The unified method simultaneously solves the power flow equations of both AC and DC systems, while the sequential method handles them through iteration. The sequential method is convenient to integrate the DC system calculation into existing AC power flow calculation programs with a little modification. Thus, the sequential method is commonly used in the practical EMS for AC/DC power systems [
22].
All the methods in the aforementioned reference use the Newton–Raphson (NR) approach to solve the non-linearity of the power flow problem. In practical applications, the NR method may not guarantee fully reliable convergence due to reasons such as unknown convergence regions, potential divergence, and the critical role of initial guess selection, where an inappropriate choice can lead to false solutions [
23]. A more robust power flow calculation method should be proposed for steady-state operation evaluations, contingency analysis, and planning for AC/DC systems.
The holomorphic embedding (HE) power flow method introduces a non-iterative approach to power flow analysis [
24]. This method reliably yields a power flow solution whenever one exists. Conversely, in scenarios where a solution is non-existent, it issues a collapse signal. The HE method has been used in power flow calculation [
25,
26,
27,
28], probabilistic power flow calculation [
29], contingency analysis [
30,
31], voltage stability margin calculation [
32,
33], dynamic simulation [
34], etc. Reference [
23] uses HE formulation for LCC-HVDC power flow calculation and obtaining reliable power flow results. Reference [
35] utilizes the HE method to solve power flow problems for VSC-HVDC power systems, considering multi-terminal configurations. However, the application of the HE method in LCC-MMC hybrid HVDC systems is currently limited, highlighting the need for the development of the HE method specifically for hybrid HVDC power flow calculations.
This paper proposes an HE-based power flow calculation method for an AC/DC power system that includes LCC-MMC hybrid HVDC. The main contributions of this paper can be summarized as follows:
- (1)
To the best of our knowledge, this paper is the first to derive the HE formulation for nonlinear power flow equations involving LCC-MMC hybrid HVDC configurations; based on the HE formulation, the germ solution and high-order power series solution process is developed; to circumvent the complexity associated with the trigonometric functions in LCC-MMC equations, this paper introduces substitute variables that simplify the HE formulation; the proposed HE formulation is easy to expand to different control modes; furthermore, the proposed HE formulation is designed to be easily adaptable to various control modes;
- (2)
To verify the availability of the HE method, the sequential solution flow chart has been constructed; this flow chart can be conveniently compatible with algorithms for AC power flow; this will make the proposed HE method more practical in the real-world large-scale system;
- (3)
Simulation results demonstrate the effectiveness of the proposed HE method, comparing accuracy and convergence performance compared with the traditional NR method.
This paper is organized as follows:
Section 2 introduces the power flow modeling and control modes for the hybrid HVDC system;
Section 3 outlines the HE formulation for LCC and MMC stations; in
Section 4, we derive the germ solution and high-order power series calculation process, including a flow chart of the sequential power flow solution process for hybrid HVDC; case studies are presented in
Section 5, and conclusions are drawn in
Section 6.
2. Modelling of Hybrid LCC-MMC HVDC System
The hybrid HVDC system includes both two-terminal and multi-terminal structures. In this paper, we construct the mathematical model for the hybrid two-terminal HVDC system.
Figure 1 illustrates the topology of this system, where one terminal employs an LCC converter and the other utilizes an MMC converter [
36]. For simplicity, the LCC station is connected to bus
i, and the MMC station is connected to bus
j. The calculation models for each converter will be elaborated on in subsequent sections.
2.1. The LCC Station Model
The structure of the LCC station is shown in
Figure 2 [
37].
In
Figure 2, the variables
Psi and
Qsi denote the active and reactive power injections from the AC side at terminal bus
i, respectively;
Pdc,i and
Qdc,i denote the active and reactive power absorbed by the converter at the DC side, respectively;
Usi represents the AC side voltage magnitude at bus
i;
Xlcc is the equivalent reactance of the converter station;
kTl represents the ratio of the converter transformer;
Idc,i and
Udc,i correspond to the current and voltage at the DC side, respectively. Assuming that the converter losses are negligible, the active and reactive power on both the AC and DC sides of the converter are approximately equal.
The power flow equations for the LCC station are provided by (1)–(3) as follows:
where
θi is the converter commutation angle (i.e., rectifier commutation angle or inverter extinct angle);
kγ is the parameter obtained by taking into account the commutation effect, usually 0.995; and
φi is the converter power factor angle.
The DC network equation for the LCC is shown in (3) as follows:
where
nc represents the number of converter stations in the DC network, with
nc equalling 2 in the context of the two-terminal HVDC system;
gij represents the admittance matrix element of the DC network; and
Nj denotes the number of converter bridges at
j.
The variables for each LCC station include
Udc,i,
Idc,i, cos
θi, cos
φi, and
kTl. To ensure that the number of power flow equations matches the number of variables, each LCC station must add two additional equations. This can be achieved by specifying two variables at predetermined values, which are also known as control modes. For the sake of generality, these two equations are not explicitly detailed, as shown in (4)–(5).
For instance, if an LCC station employs constant current control and constant transformer ratio control, (4)–(5) can be explicitly expressed as Δ
d4 = Idc,i −
= 0 and Δ
d5 =
kTl −
= 0. Here,
and
are the predetermined values. The detailed control modes for the LCC station are presented in
Section 2.3.
2.2. The MMC Station Model
The MMC is characterized by its lower frequency and reduced losses, and it does not require the addition of a filter. The structure of the MMC station is shown in
Figure 3 [
38].
In
Figure 3,
Psj and
Qsj represent the active and reactive power injections from the AC side at terminal bus
j, respectively;
Xmmc denotes the equivalent reactance of the converter;
Rmmc represents the equivalent resistance, which is composed of converter losses;
Usj represents the voltage magnitude at the AC side at terminal bus
j;
Ucj represents the voltage of the commutation bridge at the DC side;
Udc,j stands for DC side voltage; and
Idc,j represents the DC side current.
The power flow equations for the MMC station are presented in (6)–(9) as follows:
where
Mj denotes the modulation ratio, with a range from 0 to 1;
μd represents the DC voltage utilization, which changes depending on the specific modulation mode used, and is equal to
in this paper. The difference in phase angle is given by
δj =
δsj −
δcj. The parameter
Yj is defined as
Yj =
, and
βj is calculated as arctan(
Xmmc/Rmmc).
The DC network equation for the MMC is shown in (9).
The variables for each MMC station include
Udc,j,
Idc,j,
Mj,
δj,
Psj, and
Qsj. To ensure that the number of power flow equations matches the number of variables, each MMC station must add two additional equations. This can be achieved by specifying two variables at predetermined values, which are also known as control modes. For the sake of generality, these two equations are not explicitly detailed, as shown in (10)–(11).
For instance, if an MMC station employs constant voltage control and constant reactive control, (10)–(11) can be explicitly expressed as Δ
d10 = Udc,j−
= 0 and Δ
d5 =
Qsj −
= 0. Here,
and
are the predetermined values. The detailed control modes for the MMC station are presented in
Section 2.3.
2.3. Control Modes
The control modes for the LCC-MMC hybrid HVDC system is different from a pure LCC or pure MMC system. Generally, the sending end adopts the LCC station, and the receiving end adopts the MMC station.
The common control modes for the LCC and MMC stations are presented in
Table 1. For the stable operation of the DC network, it is essential to maintain the power balance and the voltage in a normal range. Generally, one end of the converter should be controlled by constant voltage, and the other converter should be controlled by other control modes besides constant voltage control.
Therefore, the hybrid two-terminal HVDC system offers selectable control mode combinations, including either LCC ➀ or ➁ with MMC ➂ or ➃, or LCC ➂ or ➃ with MMC ➀ or ➁.
3. HE Formulation for LCC and MMC Station
The traditional algorithm used for solving the nonlinear equations listed in
Section 2 is the NR method. But the convergence of the NR method depends on the selection of initial points and has disadvantages such as poor convergence in ill-conditional situations. To overcome the limitations of the NR method, this section introduces the HE formulation for the LCC and MMC stations.
3.1. HE Method
A holomorphic function is a complex-value analytic function that is infinitely complex differentiable around every point within its domain. The main property of a holomorphic function is that it can be represented by its Taylor series as a function of the complex parameter
α [
24]. For example, a holomorphic function
z(
α) can be expressed by the following:
where
z[
n] denotes the
n-th coefficient of the Taylor series expansion.
If the quantity
z to be determined is difficult to solve directly, the HE method constructs a holomorphic function
z(
α) by embedding a new complex variable
α, and embeds it into the original nonlinear equation to form an equation containing a holomorphic function that is as follows:
By performing a Taylor power series expansion on the function
z(
α), solving its power series coefficient
z[
n], and substituting it into Equation (12), the implicit function
z(
α) can be converted into an explicit function with a specific expression. For detailed information on the HE method, readers can refer to [
24].
The execution of HE-based power flow calculation involves the following steps:
Step1: Construct HE formulation for all equations;
Step2: Obtain the germ solution;
Step3: Calculate n-th power series coefficients and the target value at α = 1;
Step4: Check convergence; if convergence criterion is met or maximum iteration is reached, go to the next step; otherwise, go to Step 3;
Step 5: Output the power flow calculation results.
In this paper, we design HE formulations for both the LCC station and the MMC station, as well as for various control modes.
3.2. HE Formulation for LCC Station
For each unknown variable, a holomorphic function is constructed using the complex embedding parameter α. For instance, the unknown variable Udc,i is expressed as Udc,i(α). The holomorphic functions for other variables associated with the LCC station are as follows: kTl = kTl(α), Idc,i = Idc,i(α), cosθi = A(α), cosφi = B(α).
The HE formulations for the basic equations of the LCC station are shown in (15)–(16) as follows:
where 2.7 represents the approximate value of
. The HE formulation of the DC network equation is presented in (17).
3.3. HE Formulation for MMC Station
Similar to the LCC station, the HE formulation for MMC Equations (6)–(9) has been developed. The variables associated with the MMC station are also represented as holomorphic functions. The holomorphic functions for variables associated with the MMC station are as follows: Psj = Psj(α), Qsj = Qsj(α), Udc,j = Udc,j(α), Mj = Mj(α), cosδi = C(α), sinδi = D(α), Idc,i = Idc,i(α), cosθi = A(α), and cosφi = B(α).
The HE formulation for the power flow equations of the MMC station is presented in (18)–(20).
In (20), the parameter
λ can be calculated by the following Equation (21):
where
Udc,j[0]
Idc,j[0], and
Mj[0] are the germ solution and the value can be found in (29)–(40).
The HE formulation for the DC network equation is shown in (22).
To avoid handling trigonometric functions of cos(
δj +
βj) in (6) and sin(
δj +
βj) in (7), it is challenging to design the HE formulation directly for
δj. Therefore, we use
C = cos(
δj) and
D = sin(
δj) as variables to circumvent the need for a direct HE formulation for
δj. Once the values of
C and
D are obtained, we can recover the value of
δj. Consequently, an auxiliary equation is introduced, with its corresponding HE formulation presented in (23).
3.4. HE Formulation for Control Modes
As shown in
Table 1, there are many different control modes for LCC and MMC stations. In this paper, the LCC station uses constant current and constant transformer ratio control, whereas the MMC station employs constant voltage and reactive power control to validate the proposed method.
The HE formulation for the LCC control mode equations is shown in (24)–(25) as follows:
where
and
represent the control parameters, specifically the constant current and the constant transformer ratio.
Idc0 represents the initial DC current value, which is established by the subsequent guideline. If the LCC station employs constant current control, then
Idc0 can be set to
. If the LCC station adopts other control modes,
Idc0 can be specified with any appropriate value.
The HE formulation for the MMC control mode equations is shown in (26)–(27) as follows:
where
and
represent the control parameters of the specified MMC, namely DC voltage and reactive power, respectively;
Udc0 and
Qsj0 denote their initial values, with
Udc0 set near the standard unit value and
Qsj0 defined as
Qsj[0], which can be calculated by (38).
6. Conclusions
This paper proposes a HE-based power flow calculation method for AC/DC power systems with LCC-MMC hybrid HVDC. The HE formulation for additional nonlinear equations of LCC-MMC is designed. Subsequently, the germ solution and the high-order power series coefficient calculation process are derived. Through tests conducted on the modified IEEE 14-bus, IEEE 57-bus and 2383-bus system, the efficacy of the proposed method is demonstrated by comparing it with the traditional NR method. Compared with the traditional NR method, the HE-based method is theoretically more reliable. The major conclusions can be summarized as follows:
- (1)
The proposed HE formulation for the LCC-MMC hybrid HVDC system demonstrates practicality and accuracy when compared with the iterative NR method;
- (2)
In relatively large systems, the proposed method achieves higher computational efficiency compared to the traditional NR method;
- (3)
The proposed HE formulation can adapt to different control modes, simplifying the calculation process of the germ solution under varying control models;
- (4)
The sequential method can be implemented with minimal modifications to the existing AC power flow programs, regardless of whether using the NR or HE method; as a result, the sequential method may be more widely accepted due to practical considerations and computational costs.
The work carried out in this paper provides guidance for further research on power flow calculation based on the HE method in AC/DC systems. The proposed method has the potential to be extended to systems with multi-terminal the LCC-MMC hybrid HVDC system in future work. In new-type power systems, power flow calculations are integral to tasks such as static security analysis, power system optimal planning, probabilistic power flow calculation, and other critical functions. Furthermore, the significance of online assessment software is expected to grow, particularly in safeguarding high renewable energy-integrated power systems. Consequently, there is an urgent demand for a power flow algorithm that is both fast and reliable. Compared to traditional iterative methods, the HE method demonstrates considerable advantages in addressing these needs.