Low-Frequency AC Multiport Asynchronous Grid Connection System to Optimize Generation Costs and Mitigate Bottlenecks

Abstract

Renewable energy sources continue to increase due to the energy transition; thus, the generation output of conventional power sources is decreasing. The installation of renewable energy sources can lead to the concentration of these sources depending on geographical characteristics, which may cause bottlenecks between the renewable energy generation sites and load centers, and such bottlenecks can result in power shortages in load centers and may cause issues that limit the integration of renewable energy. Thus, this paper proposes the application of an LFAC multiport asynchronous grid connection system to solve these problems, where the frequency conversion device uses a variable frequency transformer (VFT). In addition, the installation location of the proposed LFAC multiport asynchronous grid connection system is selected using a grid partitioning technique, and the optimal power flow is performed to minimize the generation costs. The grid partitioning was conducted using the IEEE 39 bus system, and the feasibility of the proposed LFAC multiport asynchronous grid connection system was verified through simulation tests of the generation and load demands in the grid. In addition, the VFT control and optimal power flow control performances were confirmed through MATLAB/Simulink.

1. Introduction

The share of renewable energy sources has been increasing continuously due to the recent energy transition. This change is occurring as conventional power generation facilities are being converted to renewable energy sources, and new renewable energy sources are being installed. Renewable energy sources can be concentrated in specific regions depending on the geographical characteristics of each region, which can incur bottlenecks in the power transmission paths between the generation sites and the load centers. These bottlenecks result in power shortages in the load centers and limit the integration of renewable energy [1,2].

As a primary technology for integrating renewable energy, HVDC multiport grid connection technology has been widely used. This technology enables large-capacity, long-distance power transmission, thereby enhancing the flexibility of the power grid [3,4,5,6,7]. However, HVDC has the disadvantages of high initial installation costs and power conversion losses when connected to conventional AC power grids.

To address these challenges, this paper proposes the application of low-frequency AC (LFAC) multiport asynchronous grid connection technology. LFAC power transmission technology operates at a frequency lower than the standard frequency, typically around 20 Hz, and is used to link asynchronous systems. This technology allows for approximately three times the power transmission capacity compared with conventional 60 Hz lines, reduces power loss through decreased reactance, and is advantageous for long-distance subsea power transmission. Moreover, since LFAC uses AC, it is easier to connect with existing AC power grids, and it offers superior economic benefits for medium-distance power transmission compared with both HVDC and HVAC [8,9,10,11].

Previous studies have discussed the application of the LFAC system to large-scale offshore wind farms, focusing on reducing the complexity of HVDC systems, lowering investment costs, and improving wind energy economics [8,9]. Other studies have investigated optimal frequency selection to increase transmission network capacity and resolve bottlenecks [10,11]. However, while these studies primarily focused on the economic feasibility of LFAC system installation in terms of investment costs, they did not sufficiently consider minimizing generation costs based on regional load demands or selecting appropriate installation locations for the LFAC system.

Thus, in the current study, we utilized the LFAC multiport asynchronous connection technology to address bottlenecks in the power transmission paths between the generation regions and the load centers. First, a grid-partitioning method is proposed to select the installation location for the LFAC multiport asynchronous system, and optimal power flow (OPF) control is performed to minimize the generation costs for each partitioned region. Here, both generation and load simulations were conducted using the optimal power control; thus, the feasibility of the existing transmission lines and the LFAC multiport asynchronous grid connection system was confirmed, and the system’s performance was verified using MATLAB R2022b/Simulink.

2. LFAC Power Transmission Technology

2.1. LFAC System Overview

LFAC system operates at 20 Hz, which is lower than the standard frequency, and is connected to the AC system. To increase the effective power P of the AC power transmission system, the voltage magnitude must be increased or the impedance must be reduced. If the voltage at the transmitter and receiver ends is fixed, the effective power can be increased by reducing the impedance of the transmission line [10].

The basic concept of an LFAC transmission system is shown in Figure 1. The LFAC power transmission system connects existing AC power grids with low-frequency AC. The frequency converter, which converts standard AC frequencies to low frequencies, connects the two systems at both ends. There are two methods for the frequency conversion, i.e., indirect AC–DC–AC and direct AC–AC. The indirect AC–DC–AC method can utilize HVDC systems, and the direct AC–AC method includes either cycloconverters or variable frequency transformer (VFT) systems [11].

The AC–DC–AC converter can be considered a variation of the HVDC. Here, the DC transmission line is reduced to a DC bus, and the rectifier and inverter are connected in a back-to-back manner at one station to achieve frequency conversion. This method can also be used when connecting two asynchronous AC power grids. The AC–AC cycloconverter has been discussed in various applications, e.g., high-power motor drives for industrial and railway systems; however, it requires large reactive power consumption, and large filters must be designed to improve output quality [11]. In contrast, the AC–AC VFT system employs a rotating transformer with a rotor to adjust the frequency between the two systems, thereby facilitating synchronization and enabling a stable power supply. Thus, the VFT is used in this study as the frequency converter.

The VFT, as a rotary machine, possesses inertia and can efficiently perform frequency conversion and bidirectional power flow control through relatively simple DC motor control. Additionally, it offers the advantages of easy capacity expansion through parallel connections and lower investment costs.

2.2. VFT Operating Principle and Modeling

The VFT is a bidirectional power device that is capable of asynchronous grid connection and configured in the form of a wound-rotor induction motor. As shown in Figure 2, the main components of the VFT are a rotating transformer, a motor drive, and a collector system [12].

Here, the rotating transformer is used to exchange power, and the motor drive is mechanically connected to the transformer to apply torque. In addition, the motor drive adjusts the relative position between the stator and rotor, thereby controlling the amount and direction of the power transmission. The collector system transmits the current between the rotor windings and the bus.

The operating principle of the VFT is illustrated in Figure 3. When Grid A is connected to the stator side, the drive motor rotates at a speed that controls the frequency of Grid B, which is connected to the rotor side, thereby enabling asynchronous grid connection. The power output on the rotor side is relative to the power on the stator side, as well as the rotor’s rotational speed and direction [13]. For example, if the frequency of Grid B fluctuates to 55 Hz, the drive motor rotates at 5 Hz. When the frequencies of Grids A and B are both 60 Hz, the drive motor remains stationary, and the system operates similarly to a conventional transformer.

Figure 4 shows a schematic diagram of the grid connection using the VFT, where Grid A is connected to the stator, and Grid B is connected to the rotor, thereby enabling grid interconnection. The output power through the VFT can be represented by the power from both the rotor and the stator, as well as the motor drive, and can be expressed as follows.

$${\mathrm{P}}_{\mathrm{D}}={\mathrm{P}}_{\mathrm{s}}+{\mathrm{P}}_{\mathrm{r}}$$

(1)

Here, P_D denotes the mechanical output through the torque of the motor drive, P_s denotes the stator power, and P_r denotes the rotor power. The windings of the stator and rotor are linked by the same magnetic flux; however, the voltage changes proportionally with the frequency variations. This relationship can be expressed by Equation (2).

$${\mathrm{V}}_{\mathrm{r}}/{\mathrm{N}}_{\mathrm{r}}={\mathrm{V}}_{\mathrm{s}}/{\mathrm{N}}_{\mathrm{s}}\times {\mathrm{f}}_{\mathrm{r}}/{\mathrm{f}}_{\mathrm{s}}$$

(2)

Here, N_s and f_s denote the stator windings and stator frequency, and air-gap flux, respectively, and N_r and f_r denote the rotor windings and frequency, respectively. Note that the rotor speed is proportional to the difference between the stator and rotor frequencies.

$${\mathrm{f}}_{\mathrm{m}}={\mathrm{f}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{r}}$$

(3)

$${\mathsf{\omega }}_{\mathrm{m}}={\mathrm{f}}_{\mathrm{m}}\times 120/\mathrm{P}$$

(4)

Here, f_m denotes the rotor frequency speed, ω_m denotes the angular speed of the rotor, and P is the number of poles in the VFT. When power is transferred from the stator to the rotor, the drive power of the motor can be expressed as follows.

$${\mathrm{P}}_{\mathrm{D}}={\mathrm{V}}_{\mathrm{r}}{\mathrm{I}}_{\mathrm{r}}(1-{\mathrm{f}}_{\mathrm{s}}/{\mathrm{f}}_{\mathrm{r}})={\mathrm{P}}_{\mathrm{r}}(1-{\mathrm{f}}_{\mathrm{s}}/{\mathrm{f}}_{\mathrm{r}})$$

(5)

Here, I_r denotes the magnitude of the rotor current. The rotor power is proportional to the mechanical output of the motor drive and is influenced by both the stator and rotor frequencies. Thus, the driving torque of the motor drive can be expressed as follows.

$${\mathrm{T}}_{\mathrm{D}}={\mathrm{P}}_{\mathrm{D}}/{\mathrm{f}}_{\mathrm{m}}={\mathrm{N}}_{\mathrm{r}}{\mathsf{\Psi }}_{\mathrm{a}}{\mathrm{I}}_{\mathrm{r}}$$

(6)

When a VFT is used for power transmission, it operates while maintaining a nearly constant magnetic flux; thus, the magnitude of the driving torque is proportional to only the rotor current.

2.3. VFT Frequency Conversion Controller Structure

Figure 5 shows the control block diagram of the frequency conversion system. As can be seen, the system is partitioned into a speed controller and a phase controller. The phase and frequency of the three-phase voltages on both the primary and secondary sides can be detected using a PLL. After detection, the frequency of the primary and secondary sides is input to the speed controller to obtain the speed control command value τ_f*, and the phases of the primary and secondary sides are input to the phase controller to obtain the phase control command value τ_θ*. Then, these two command values are input as torque command values for the motor drive of the VFT, which performs speed and phase control based on the command values.

In this study, a DC motor was used as the drive motor for the VFT, and the corresponding structure of the speed controller is shown in Figure 6. Here, the current controller is positioned in the inner loop, and the speed controller is used in the outer loop. PI controller was used for the speed control of the VFT, and the transfer function of the PI speed controller can be expressed as follows [13].

$${\mathrm{K}}_{\mathrm{ps}}={\displaystyle \frac{\mathrm{J}{\omega }_{\mathrm{cs}}}{{\mathrm{K}}_{\mathrm{T}}}}$$

(7)

$${\mathrm{K}}_{\mathrm{is}}={\displaystyle \frac{\mathrm{J}{\omega }_{\mathrm{cs}}^2}{5{\mathrm{K}}_{\mathrm{T}}}}$$

(8)

As the VFT performs speed control, phase changes occur, and if the phase of the secondary side is not synchronized with the phases of the primary side and the rotating transformer, synchronization forces may occur; thus, making phase control is essential. As shown in Figure 5, the voltage of the primary system connected to the stator windings and the secondary system connected to the rotor windings was measured, and the phase was detected using a PLL. The rotating transformer compensates for the phase difference accordingly, and the relationship between the phases can be expressed as follows.

$${\theta }_{\mathrm{VFT}}={\theta }_1-{\theta }_2$$

(9)

Here, θ_VFT is the phase angle that must be compensated by the VFT. Figure 7 shows the block diagram of the VFT phase controller. Based on the reference value for phase compensation, a PI controller is employed to add the phase synchronization command T_θ* for the primary and secondary systems, which is then injected into the drive system, thereby allowing synchronization of both the frequency and phase between the primary and secondary systems.

2.4. Design of the LFAC Multiport Asynchronous Grid-Connected System Reference Model

Figure 8 shows a schematic diagram of the LFAC multiport grid-connected system using VFTs, where Grids A, B, and C are connected via VFTs, and the system is interconnected in a multiport configuration.

The LFAC system connected to Bus A–B, Bus B–C, and Bus A–C has the same structural design, and the discussion focuses on the interconnection between Grids A and B. The reference model configuration for the LFAC multiport asynchronous grid-connected system is shown in Figure 9.

Figure 9 proposes a reference model for a multiport grid-connected system, incorporating a VFT as the frequency converter, with the corresponding parameter values for the reference model provided in

Table 1. LFAC multiport grid-connected system reference model parameters.

Component	Parameter
VFT (A side)	100 MW $\times$ 6, 22.9 kV
Capacitor bank (A side)	25 MVAR $\times$ 6
Transformer (A side)	300 MW, 22.9 kV/154 kV
DC motor (A side)	3000 HP $\times$ 6
VFT (B side)	100 MW $\times$ 6, 22.9 kV
Capacitor bank (B side)	25 MVAR $\times$ 6
Transformer (B side)	300 MW, 22.9 kV/154 kV
DC motor (B side)	3000 HP $\times$ 6
LFAC system circuit breakers	30 kA
LFAC transmission line	154 kV, 0.047 Ω/km,

. The equipment connected to Grid A is based on a multiport connection. Each VFT has a rated capacity of 100 MW [12,14,15], and six units are connected in parallel, resulting in a total capacity of 600 MW. The DC motor connected to each VFT is rated at 3000 HP [12,14]. The transformer has a capacity of 300 MW and is a high-voltage transformer designed for operation at 20 Hz. According to the Korea Electric Power Corporation’s regulations for the use of transmission and distribution facilities, the LFAC transmission line is designed with a voltage level of 154 kV and two circuits to achieve a total transmission capacity of 300 MW (based on a standard transmission capacity of 150 MW per circuit). The circuit breaker rating is calculated based on the three-phase short-circuit current, which is generally higher than the calculated value in typical generation-to-load systems. In cases where the power system includes both generation sources and loads, the fault current varies depending on the fault location; thus, manual calculations are challenging. In addition, the grounding impedance may vary depending on the season and weather conditions. Here, fault currents of approximately five to seven times the normal current can occur in the event of a fault on the connected lines; thus, the circuit breaker is rated at 30 kA.

3. Selection of LFAC System Installation Location Using Grid Partitioning Method

The grid partitioning method is typically based on the operator’s experience, often disregarding the internal structural connections. To prevent this, a graph model is constructed to represent the relationships between busses. Each bus is denoted as v_i and the transmission line between busses as e_ij, simplifying the system into an undirected graph G = (V,E), where V is the set of bus v_i, and E is the set of transmission lines e_ij. The graph adjacency matrix W and the degree matrix D are expressed as follows [16,17].

$${\mathrm{w}}_{\mathrm{ij}}=\left\{\begin{array}{cc}1& {\mathrm{e}}_{\mathrm{ij}}\in \mathrm{E}\\ 0& {\mathrm{e}}_{\mathrm{i}}\notin \mathrm{E}\end{array}\right.$$

(10)

$${\mathrm{d}}_{\mathrm{ij}}=\left\{\begin{array}{cc}{\displaystyle \sum _{\mathrm{j}}}{\mathrm{w}}_{\mathrm{ij}}& \mathrm{i}=\mathrm{j}\\ 0& \mathrm{i}\ne \mathrm{j}\end{array}\right.$$

(11)

Here, w_ij is the matrix element of the adjacency matrix W, and d_ij is the matrix element of the degree matrix D. In an undirected graph, the adjacency matrix is w_ij = w_ji. Figure 10 shows a schematic diagram of the graph theory.

The spectral clustering algorithm is partitioned into unnormalized clustering and normalized spectral clustering, and the difference between the two lies in the degree of normalization of the Laplacian matrix. In normalized spectral clustering, a symmetric Laplacian matrix is utilized to analyze the relationships between the data effectively, thereby allowing for more precise cluster separation. The normalized Laplacian clustering algorithm is described as follows.

Input: n sample points $\mathrm{X}=\{{\mathrm{x}}_1,{\mathrm{x}}_2, \cdots , {\mathrm{x}}_{\mathrm{n}}\}$, number of clusters $\mathrm{k}$.

• Calculate the n × n similarity matrix S.
• Calculate the degree matrix D.
• Compute the normalized Laplacian matrix ${\mathrm{L}}_{\mathrm{sym}}$.
• ${\mathrm{L}}_{\mathrm{sym}}={\mathrm{D}}^{-1/2}\mathrm{L}{\mathrm{D}}^{-1/2}(\mathrm{D}-\mathrm{S}){\mathrm{D}}^{-1/2}$.
• Calculate the eigenvalues of ${\mathrm{L}}_{\mathrm{sym}}$ and obtain the first $\mathrm{k}$ eigenvectors of the matrix ${\mathrm{L}}_{\mathrm{rw}}$ as columns ${\mathrm{u}}_1,{\mathrm{u}}_2,\cdots {\mathrm{u}}_{\mathrm{k}}$.
• Form the matrix $\mathrm{U}=\{{\mathrm{u}}_1,{\mathrm{u}}_2,\cdots {\mathrm{u}}_{\mathrm{k}}\}$, where $\mathrm{U}\notin {\mathrm{R}}^{\mathrm{n}\times \mathrm{k}}$.
• ${\mathrm{y}}_{\mathrm{i}}\in {\mathrm{R}}^{\mathrm{k}}$ is the vector in the $\mathrm{i}$-th row of $\mathrm{U}$ for $\mathrm{i}=1,2,\cdots ,\mathrm{n}$.
• Normalize ${\mathrm{y}}_{\mathrm{i}}\in {\mathrm{R}}^{\mathrm{k}}$ so that $\left|{\mathrm{y}}_{\mathrm{i}}^{*}\right|=1$.
• Use the K-means algorithm to cluster the new sample points $\mathrm{Y}=\{{\mathrm{y}}_1^{*},{\mathrm{y}}_2^{*},\cdots ,{\mathrm{y}}_{\mathrm{n}}^{*}\}$ into clusters ${\mathrm{C}}_1,{\mathrm{C}}_2,\cdots ,{\mathrm{C}}_{\mathrm{k}}$.

Output: Clusters ${\mathrm{A}}_1,{\mathrm{A}}_2,\cdots , {\mathrm{A}}_{\mathrm{n}}$. ${\mathrm{A}}_{\mathrm{i}}=\left\{\mathrm{j}\right|{\mathrm{y}}_{\mathrm{i}}\in {\mathrm{C}}_{\mathrm{i}}\}$.

Here, x_n denotes the data point representing v_i. By applying the electrical distance as a weight to the adjacency matrix, the similarity s_ij = s(v_i,v_j) can be calculated, and the similarity matrix S_ij = (s_ij), where i,j = 1,⋯, n can be represented. In this study, the input data comprise 39 samples using the IEEE 39 bus system, and k is set to 3.

In the power grid, the weight of a line represents the similarity or distance between busses, and, in this study, the voltage sensitivity is selected as the electrical distance. The voltage sensitivity can be obtained through the Jacobian matrix and is expressed as follows.

$$\left[\begin{array}{c}△\mathrm{P}\\ △\mathrm{Q}\end{array}\right]=\left[\begin{array}{cc}{\mathrm{J}}_{11}& {\mathrm{J}}_{12}\\ {\mathrm{J}}_{21}& {\mathrm{J}}_{22}\end{array}\right]\left[\begin{array}{c}△\theta \\ △\mathrm{V}\end{array}\right]$$

(12)

Here, P, Q, θ, and V denote the active power, reactive power, phase angle, and voltage magnitude, respectively. Note that the voltage sensitivity can be obtained from J₂₂; thus, the electrical distance between the i-th and j-th busses can be expressed as follows.

$${\mathrm{e}}_{\mathrm{i}}{,}_{\mathrm{j}}={\displaystyle \frac{{\left({\mathrm{J}}_{22}\right)}_{\mathrm{i},\mathrm{j}}+{\left({\mathrm{J}}_{22}\right)}_{\mathrm{j},\mathrm{i}}}{2}}$$

(13)

When the impedance of a transmission line is very small, the electrical distance between two busses generally becomes large, which indicates that the connection between the busses is very close.

Table 2. Electrical distance between busses.

Bus	Electrical Distance	Bus	Electrical Distance
1–2	2.2370	14–15	3.9052
1–39	1.7099	15–16	10.5872
2–3	6.1105	16–17	9.3774
2–25	63.1136	16–19	4.5170
2–30	2.5000	16–21	4.6583
3–4	2.9555	16–24	9.2128
3–18	6.5669	17–18	11.0252
4–5	4.9133	17–27	4.6383
4–14	4.8669	19–20	3.5983
5–6	29.8290	19–33	6.2514
5–8	6.3653	20–34	4.9842
6–7	7.0995	21–22	5.9826
6–11	10.5562	22–23	7.1177
6–31	6.6867	22–35	6.5000
7–8	18.6908	23–24	3.5124
8–9	1.8011	23–36	5.5797
9–39	1.7084	25–26	3.3809
10–11	22.1207	25–37	5.3967
10–13	22.1542	26–27	7.0132
10–32	6.5000	26–28	2.0947
11–12	0.8513	26–29	1.9152
12–13	0.8526	28–29	6.7071
13–14	8.9889	29–38	8.2671

shows the electrical distances between busses calculated using the voltage sensitivity, and

Table 3. Busses by partitioned region.

Clustering	Bus
Region 1	1, 2, 3, 18, 25, 26, 27, 28, 29, 30, 37, 38, 39
Region 2	4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 32
Region 3	15, 16, 17, 19, 20, 21, 22, 23, 24, 33, 34, 35, 36

shows the results of spectral clustering using voltage sensitivity as a weight, which partitioned the 39 busses into three regions. Thus, the installation locations of the LFAC multiport asynchronous grid-connected system can be selected as transmission lines between the partitioned regions. Figure 11 shows the possible installation locations of the LFAC multiport asynchronous grid-connected system (marked as Points A, B, C, D, and E) in the IEEE 39 bus system.

4. Optimal Power Flow Control for Minimizing Generation Costs

4.1. Optimal Power Flow Control for LFAC Multi-Port System

The advantage of the economic dispatch formula to select the minimum cost for the generation units is that it can be calculated quickly and solved reasonably. However, two major disadvantages must be considered. First, it is somewhat inaccurate because it approximates losses and assumes that the transmission capacity is infinite. Second, it does not provide information about the line flows.

The OPF compensates for these disadvantages by replacing a single power balance equation with node power flow equations. Here, the generator limits are maintained as inequality constraints, and additional constraints are applied to the line flows. Linear optimization problems can be solved effectively using linear programing (LP); thus, this paper selects LP for OPF [18,19].

The linearized OPF formula can be expressed as follows.

$$\mathrm{x}=\left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g}}\\ {\mathrm{P}}_{\mathrm{B}}\\ \theta \end{array}\right]$$

(14)

Here, P_g is the vector representing the generation increments for all power plants k as [P_g1 P_g2… P_gk ]^T, which indicates the amount of power produced by each power plant. In addition, P_B is the vector representing the line flows as [P_b1 P_b2 … P_bM ]^T, where M is the number of lines and represents the amount of power flow passing through each transmission line. Finally, θ denotes the vector representing the bus angles (in radians) [θ₁ θ₂ … θ_N], where N is the number of buses and represents the voltage angle of each bus.

Note that all equality constraints are expressed in a single matrix relationship, and the equality constraints arising from line flows and injections are given in Equations (15) and (16). In addition, the inequality constraints are given in Equation (17).

$${\mathrm{P}}_{\mathrm{B}}=\left[\mathrm{D}\times \mathrm{A}\right]\times \theta$$

(15)

$$\mathrm{P}={\mathrm{B}}^{’}\theta$$

(16)

$$\left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g},\min }\\ -{\mathrm{P}}_{\mathrm{B},\max }\\ -\pi \end{array}\right]\le \left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g}}\\ {\mathrm{P}}_{\mathrm{B}}\\ \theta \end{array}\right]\le \left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g},\max }\\ {\mathrm{P}}_{\mathrm{B},\max }\\ \pi \end{array}\right]$$

(17)

Here, D is the diagonal matrix representing the impedance of the transmission lines. A is the bus–line incidence matrix, which represents the connections between busses and lines in the power system, which shows the flow of power. In addition, P is the power injection vector, and B’ is the transposed susceptance matrix, representing the admittance matrix of the transmission lines.

4.2. Optimal Power Control for LFAC Multiport Asynchronous Grid-Connected System

The OPF is crucial in terms of ensuring the stable and economic operation of power systems. To achieve this, the generator outputs must be minimized according to the operational and planning requirements of the power system. The OPF determines the output commands for the generators in consideration of several constraints, e.g., the active power of each generator and the load conditions [18]. In this study, the OPF was applied to the LFAC multiport asynchronous grid system to determine the output commands of the generators connected to the frequency converter. Figure 12 shows the single-line diagram of the LFAC multiport asynchronous grid-connected system for the OPF based on the partitioning of the IEEE 39 bus system. The generation and load capacities of each region were selected according to the grid configuration, and the admittance of the transmission lines was selected based on the low frequency, which is primarily determined by the inductance of the lines.

$$\mathrm{Cos}\mathrm{t} \mathrm{Function} =\min (\mathrm{s}1{\mathrm{P}}_{\mathrm{g}1}+\mathrm{s}2{\mathrm{P}}_{\mathrm{g}2}+\mathrm{s}3{\mathrm{P}}_{\mathrm{g}3})$$

(18)

Equation (18) represents the objective of the multiport asynchronous system, where s1, s2, and s3 represent the generation costs for each type of generator and P_g1, P_g2, and P_g3 represent the generation amounts for Regions A, B, and C, respectively. Note that generation costs for each generator type are based on 2024 data from the Korea Power Exchange’s power statistics information system (

Table 4. Generation costs per hour by type of generator.

Power Generation Type	Cost
s1 (diesel generation)	37,450,000 KRW/puMWhr
s2 (wind turbine generation)	12,660,000 KRW/puMWhr
s3 (wind turbine generation)	12,660,000 KRW/puMWhr

) [20]. In addition, the power supply for the three regions is assumed to come from wind-power generation.

The loads of buses 1, 2, and 3 are 2620 MW, 1330 MW, and 2350 MW, respectively, as shown in

Table 5. Hourly generation capacity and load capacity by region.

Region	Generator Capacity [MWh]		Load [MWh]
1	max	4292	2620
	min	1073
2	max	1920	1330
	min	300
3	max	3240	2350
	min	810

. The pu value is expressed based on the 100 MW capacity of LFAC transmission.

The objective function is given in Equation (19), and the equality constraint calculations are given in Equations (20) to (22) using Equations (15)–(17).

$$\mathrm{Z}\left(\mathrm{x}\right)=\left[\begin{array}{ccccccccc}\mathrm{37,450,000}& \mathrm{12,660,000}& \mathrm{12,660,000}& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g}1}\\ {\mathrm{P}}_{\mathrm{g}2}\\ {\mathrm{P}}_{\mathrm{g}3}\\ {\mathrm{P}}_{\mathrm{B}12}\\ {\mathrm{P}}_{\mathrm{B}13}\\ {\mathrm{P}}_{\mathrm{B}23}\\ {\theta }_1\\ {\theta }_2\\ {\theta }_3\end{array}\right]$$

(19)

Here, P_B12, P_B13, and P_B32 represent the power flow through the transmission lines, and θ₁, θ₂, and θ₃ represent the bus voltage angles. D × A is expressed in Equation (20), and B’ can be represented by Equation (21).

$$\mathrm{D}\times \mathrm{A}=\left[\begin{array}{ccc}28.86& 0& 0\\ 0& 54.11& 0\\ 0& 0& 43.29\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 0& -1& 1\end{array}\right]=\left[\begin{array}{ccc}28.86& -28.86& 0\\ 54.11& 0& -54.11\\ 0& -43.29& 43.29\end{array}\right]$$

(20)

$${\mathrm{B}}^{’}=\left[\begin{array}{ccc}82.97& -28.86& -54.11\\ -28.86& 72.15& -43.29\\ -54.11& -43.29& 97.40\end{array}\right]$$

(21)

$${\mathrm{A}}_{\mathrm{eq}}\mathrm{x}=\left[\begin{array}{c}0\\ 0\\ 0\\ -1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ -1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ -1\end{array}\begin{array}{c}-1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ -1\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ -1\\ 0\\ 0\\ 0\end{array}\begin{array}{c}28.86\\ 54.11\\ 0\\ 82.97\\ -28.86\\ -54.11\end{array}\begin{array}{c}-28.86\\ 0\\ -43.29\\ -28.86\\ 72.15\\ -43.29\end{array}\begin{array}{c}0\\ -54.11\\ 43.29\\ -54.11\\ -43.29\\ 97.40\end{array}\right]\left[\begin{array}{c}{\mathrm{P}}_{\mathrm{g}1}\\ {\mathrm{P}}_{\mathrm{g}2}\\ {\mathrm{P}}_{\mathrm{g}4}\\ {\mathrm{P}}_{\mathrm{B}12}\\ {\mathrm{P}}_{\mathrm{B}13}\\ {\mathrm{P}}_{\mathrm{B}23}\\ {\theta }_1\\ {\theta }_2\\ {\theta }_3\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ -26.2\\ -13.3\\ -23.5\end{array}\right]$$

(22)

$$\left[\begin{array}{c}\begin{array}{c}10.73\\ 0.30\\ 8.10\end{array}\\ \begin{array}{c}-3\\ -3\\ -3\end{array}\\ \begin{array}{c}-\pi \\ -\pi \\ -\pi \end{array}\end{array}\right]\le \left[\begin{array}{c}\begin{array}{c}{\mathrm{P}}_{\mathrm{g}1}\\ {\mathrm{P}}_{\mathrm{g}2}\\ {\mathrm{P}}_{\mathrm{g}3}\end{array}\\ \begin{array}{c}{\mathrm{P}}_{\mathrm{B}12}\\ {\mathrm{P}}_{\mathrm{B}13}\\ {\mathrm{P}}_{\mathrm{B}23}\end{array}\\ \begin{array}{c}{\theta }_1\\ {\theta }_2\\ {\theta }_3\end{array}\end{array}\right]\le \left[\begin{array}{c}\begin{array}{c}42.92\\ 19.20\\ 32.40\end{array}\\ \begin{array}{c}3\\ 3\\ 3\end{array}\\ \begin{array}{c}\pi \\ \pi \\ \pi \end{array}\end{array}\right]$$

(23)

Based on Equations (20) and (21), the matrix for equality constraints can be expressed as Equation (22), and the inequality constraints can be expressed as Equation (23). The results of the OPF for the LFAC multiport power flow optimization in the asynchronous grid-connected system are shown in

Table 6. Results of the LFAC multiport power flow optimization.

Parameter	Value
${\mathrm{P}}_{\mathrm{G}1}$	2020.00
${\mathrm{P}}_{\mathrm{G}2}$	1840.00
${\mathrm{P}}_{\mathrm{G}3}$	2440.00
${\mathrm{P}}_{\mathrm{B}12}$	−300.00
${\mathrm{P}}_{\mathrm{B}13}$	−300.00
${\mathrm{P}}_{\mathrm{B}32}$	−210.00
${\theta }_1$	−3.1416
${\theta }_2$	−3.0376
${\theta }_3$	−3.0861

(with the pu values converted back to actual values). Based on these results, the VFT can perform bidirectional power control to regulate power flow and alleviate bottlenecks.

5. Simulation

5.1. Performance Verification of VFT Using MATLAB

To verify the performance of frequency conversion and power control using the VFT, two asynchronous systems were connected to the stator and rotor windings of a doubly-fed induction motor, respectively. The frequency and phase synchronization between the low-frequency system and the conventional system were achieved by detecting the voltage at Points P2 and P3 through a PLL. Based on this, the speed reference value for the DC motor was calculated using the drive system, and the rotational axis speed of the VFT was controlled through a controller. Additionally, to exclude the effects of the initial transient state, a three-phase circuit breaker was installed between Points P2 and P3. The breaker was kept open until the system stabilized at 1 s, after which synchronization was confirmed, and the breaker was closed. Figure 13 illustrates the VFT simulation model using MATLAB/Simulink, and

Table 7. Multiport power flow optimization results by case.

Component	Parameter
Grid A	22.9 kV, 60 Hz
Grid B	22.9 kV, 20 Hz
Load 1	0–5 MW
Load 2	0–5 MW
VFT	100 MW, 22.9 kV
Tr	22.9 kV/500 V
DC motor	3000 HP

presents the VFT simulation parameters.

The rated frequency of Grid A connected to the stator windings is 60 Hz, while the rated frequency of Grid B connected to the rotor windings is 20 Hz. The circuit breaker between Points P2 and P3 was kept open until 2 s, allowing the two points to synchronize their phases, and was closed at 1 s after synchronization was confirmed. Figure 14 shows the simulation waveform of frequency synchronization using the VFT, and Figure 15 shows the simulation waveform of phase synchronization. These results confirm that the frequency synchronized to 20 Hz within 1 s, the point at which the breaker was closed, and the phase also synchronized with Grid B.

Additionally, to verify the power control performance of the VFT, power was calculated using the voltage and current measured at Point P2. Figure 16 illustrates the bidirectional power control waveform, where changes in power flow were simulated by increasing Load 2 at 3 s and Load 1 at 5 s.

5.2. Feasibility Evaluation of the LFAC Multiport Asynchronous Grid-Connected System Reference Model

When the output decreases or stops in a region due to the intermittent nature of renewable energy sources, power is supplied by either expensive diesel generators or sourced from other regions where generation costs are lower, taking into account both generation and constraint costs. However, when the capacity of the transmission lines connecting the regions reaches or approaches its limit, even low-cost power cannot be supplied directly, and power must be rerouted through other regions. If the detour transmission line is long, high transmission costs are incurred, which increases overall costs. Thus, to address this issue, the LFAC multiport system is applied to the existing transmission lines to enhance transmission capacity, thereby reducing both transmission and constraint costs. This paper evaluates the feasibility of this technique through case-by-case simulation tests. Based on the grid partitioning, the minimum and maximum capacities of the generators were selected according to regional generation and demand. The maximum and minimum values for each region’s generators are shown in Table 5. In this simulation, the three regions were assumed to use wind turbines as the power source, and power flow optimization was performed for each case. In Case 1, the power generation stops in region 1 because of the intermittent characteristics of the wind turbines, and diesel generation is used. In Case 2, diesel generation is used in region 2, and in Case 3, diesel generation is used in region 3. The results of the power flow optimization for each case are given in

Table 8. Multiport power flow optimization results by case.

Parameter	Case 1		Case 2		Case3
	HVAC	LFAC	HVAC	LFAC	HVAC	LFAC
${\mathrm{P}}_{\mathrm{G}1}$	2420.00	2020.00	2782.50	3107.50	2706.67	2880.00
${\mathrm{P}}_{\mathrm{G}2}$	1500.00	1840.00	1130.00	730.00	1443.33	1670.00
${\mathrm{P}}_{\mathrm{G}3}$	2380.00	2440.00	2387.50	2462.50	2150.00	1750.00
${\mathrm{P}}_{\mathrm{B}12}$	−100.00	−300.00	100.00	300.00	−13.00	−40.00
${\mathrm{P}}_{\mathrm{B}13}$	−100.00	−300.00	62.50	187.50	100.00	300.00
${\mathrm{P}}_{\mathrm{B}32}$	−70.00	−210.00	100.00	300.00	−100.00	−300.00
${\theta }_1$	−3.1416	−3.1416	−3.0376	−3.0376	−3.0861	−3.0861
${\theta }_2$	−3.0376	−3.0376	−3.1416	−3.1416	−3.0723	−3.0723
${\theta }_3$	−3.0861	−3.0861	−3.0723	−3.0723	−3.1416	−3.1416

.

Low frequency increases the weight of transformers. To address this issue, either existing transformers with a rated voltage and capacity three times higher can be applied, or transformers can be redesigned specifically for 20 Hz. The redesigned 20 Hz transformer is approximately 75% more expensive than the existing transformer [21]. Therefore, the cost is calculated based on the 20 Hz transformer.

The installation cost estimation was based on a capacity of 600 MW. The cost estimate for the installation of the LFAC multiport system includes three VFTs at KRW 313,447 million, three transformers at KRW 38,298 million, and six circuit breakers at KRW 20,280 million (total: KRW 372,025 million). When considering a 30-year usage period for the equipment, the annual installation cost is calculated as KRW 12,401 million. Since the transformer operates at a frequency of 20 Hz, a high-voltage transformer is required. Accordingly, the installation cost for the 20 Hz transformer is applied. Additionally, the aforementioned HVDC multiport system installation cost was estimated as follows: it includes three back-to-back converters at KRW 608,400 million, three DC circuit breakers at KRW 8081 million, and three AC circuit breakers at KRW 2808 million (total: KRW 619,289 million). When considering a 30-year usage period for the equipment, the annual installation cost is calculated as KRW 20,643 million [21,22]. The comparison of the initial installation costs of the LFAC multiport system and the HVDC multiport system is shown in

Table 9. Comparison of installation costs of LFAC and HVDC multiport systems.

	Component	Installation Costs (million KRW/MWh)	Total Cost (million KRW/MWh)
LFAC multiport system	VFT	313,447	372,025
	${\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{r}}_{20\mathrm{Hz}}$	38,298
	$\mathrm{C}\mathrm{B}$	20,280
HVDC multiport system	${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}}_{\mathrm{B}\mathrm{T}\mathrm{B}}$	608,400	619,289
	${\mathrm{C}\mathrm{B}}_{\mathrm{D}\mathrm{C}}$	8081
	${\mathrm{C}\mathrm{B}}_{\mathrm{A}\mathrm{C}}$	2808

.

5.2.1. Case 1—Diesel Generation in Region 1

In Case 1, it was assumed that when wind-power generation stops in region 1 due to its intermittent characteristics, power is supplied through diesel generation. Ideally, power should be supplied from other regions where generation costs are lower; however, the limited capacity of the existing transmission lines forces power to be generated by more expensive diesel generators in the region. The results of this case are shown in

Table 10. Comparison of hourly annual generation and installation costs in Case 1.

Parameter	HVAC Transmission Line (million KRW/MWh)	LFAC Multiport Line (million KRW/MWh)	Increase/Decrease Amount (million KRW/MWh)	Increase/Decrease Rate (%)
Generation cost	12,074,383	11,217,640	−856,742	−7.1
Installation cost	-	12,401	12,401	-
Total	12,074,383	11,230,041	−844,342	−6.99

. The annual cost of optimal power control using HVAC transmission lines is KRW 12,074,383 million. The application of the LFAC multiport system and optimal power control reduces the annual generation cost to KRW 11,23,041 million. The LFAC multiport system results in a cost reduction of 6.99% compared with the existing transmission line. Figure 17 shows the power transmission capacity for Case 1, where (a) represents the 60 Hz transmission line, and (b) represents the LFAC transmission line. The generation capacity by region is based on the power flow optimization obtained in Section 4 applied to Case 1.

5.2.2. Case 2—Diesel Generation in Region 2

In Case 2, it is assumed that power is supplied via diesel generation when wind-power generation in region 2 stops due to its intermittent characteristics. The results of this case are shown in

Table 11. Comparison of hourly annual generation and installation costs in Case 2.

Parameter	HVAC Transmission Line (million KRW/MWh)	LFAC Multiport Line (million KRW/MWh)	Increase/Decrease Amount (million KRW/MWh)	Increase/Decrease Rate (%)
Generation cost	9,311,388	8,454,646	−856,742	−9.2
Installation cost	-	12,401	12,401	-
Total	9,311,388	8,467,047	−844,342	−9.07

. The annual cost of optimal power control using HVAC transmission lines is KRW 9,311,388 million. When the LFAC multiport system and optimal power control are applied, the annual generation cost is reduced to KRW 8,467,047 million. This results in a 9.07% cost reduction compared with the existing HVAC transmission line. Figure 18 shows the power transmission capacity for Case 2, where (a) represents the 60 Hz transmission line, and (b) represents the LFAC transmission line.

5.2.3. Case 3—Diesel Generation in Region 3

In Case 3, it is assumed that power is supplied via diesel generation when wind-power generation in region 3 stops due to its intermittent characteristics. The results of this case are shown in

Table 12. Comparison of hourly annual generation and installation costs in Case 3.

Parameter	HVAC Transmission Line (million KRW/MWh)	LFAC Multiport Line (million KRW/MWh)	Increase/Decrease Amount (million KRW/MWh)	Increase/Decrease Rate (%)
Generation cost	11,496,082	10,639,339	−856,742	−7.45
Installation cost	0	12,401	12,401	-
Total	11,496,082	10,651,740	−844,342	−7.34

. The annual cost of optimal power control using HVAC transmission lines is KRW 11,496,082 million. When the LFAC multiport system and optimal power control are applied, the annual generation cost is reduced to KRW 10,641,740 million, resulting in a 7.34% cost reduction compared with the existing HVAC transmission line. Figure 19 shows the power transmission capacity for Case 3, where (a) represents the 60 Hz transmission line, and (b) represents the LFAC transmission line.

5.3. Performance Evaluation of the LFAC Multi-Port Asynchronous Grid-Connected System Reference Model

The performance of the optimal power control for the LFAC multiport asynchronous grid-connected system was evaluated using MATLAB/Simulink. This simulation is based on power flow optimization, and case transitions were performed to simulate scenarios according to the characteristics of wind turbine generation. The corresponding model is shown in Figure 20, and the relevant parameters are given in

Table 13. Parameters of the asynchronous multiport equivalent circuit.

Parameter	HVAC			LFAC
	Case 1	Case 2	Case 3	Case 1	Case 2	Case 3
PG1	154 kV, 60 Hz			154 kV, 20 Hz
PG2	154 kV, 60 Hz			154 kV, 20 Hz
PG3	154 kV, 60 Hz			154 kV, 20 Hz
Degree 1	−180.00	−174.04	−176.82	−180.00	−174.04	−176.82
Degree 2	−174.04	−180.00	−176.03	−174.04	−180.00	−176.03
Degree 3	−176.82	−176.03	−180.00	−176.82	−176.03	−180.00
Branch 1-2	60 Hz, 150 km			20 Hz, 150 km
Branch 1-3	60 Hz, 80 km			20 Hz, 80 km
Branch 3-2	60 Hz, 100 km			20 Hz, 100 km

. Here, the angle of each generator and the angle of the bus connected to it are equal.

The simulation performed case transitions every 2 s, with Case 1 represented by the orange area, Case 2 represented by the green area, and Case 3 represented by the blue area. Figure 21 shows the waveform of the power flow through the transmission lines during the transition from Case 1 to Case 2, Figure 22 shows the waveform during the transition from Case 2 to Case 3, and Figure 23 shows the waveform during the transition from Case 3 to Case 1. The results for each case confirm that the LFAC transmission lines provide three times the capacities of the conventional HVAC transmission lines. In addition, the performance of the LFAC multiport asynchronous grid-connected system reference model was verified through OPF control.

6. Conclusions

This paper has proposed the LFAC multiport asynchronous grid-connected system to optimize generation costs and resolve bottlenecks, and we assessed its feasibility. The proposed LFAC power transmission system operates at a lower frequency of 20 Hz com-pared with the conventional 60 Hz, which necessitates the use of frequency conversion equipment. In this paper, VFT, which is easy to synchronize and provides stable power supply, was selected as the frequency converter. Then, the LFAC multi-port asynchronous grid connection system model was designed.

To determine the installation locations for the LFAC multiport asynchronous grid-connected system, we conducted a partitioning process of the power grid. Here, partitioning was based on calculating the electrical distances using the voltage sensitivity, and the power system was divided into three regions by applying spectral clustering techniques. Based on these regions, we performed OPF control to minimize the generation costs. The goal of OPF control is to transfer the power produced by the generators to the load efficiently while reducing bottlenecks between the generation sites and the load centers.

Through case-by-case OPF control, we compared the installation and generation costs between the existing transmission line system and the proposed LFAC multiport asynchronous grid-connected system. The results demonstrated that the proposed LFAC system offers cost savings compared with traditional transmission line-based systems.

In addition, MATLAB/Simulink simulations were performed to evaluate the performance of the VFT and analyze the change in the active power according to the case changing. The results of this analysis confirmed that the reference model contributes to the optimization of the generation costs and alleviating bottlenecks. Thus, the proposed LFAC multiport asynchronous grid-connected system was verified to be an effective method for reducing both generation and installation costs compared to conventional 60-Hz transmission line systems.