# Superconducting Saturable Core Reactor as Variable Inductance for Controlling the Power Flow in a Transmission Line

Leonardo Miúdo, João Murta Pina, Nuno Amaro

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Leonardo Miúdo, João Murta Pina, Nuno Amaro. Superconducting Saturable Core Reactor as Variable Inductance for Controlling the Power Flow in a Transmission Line. 14th Doctoral Conference on Computing, Electrical and Industrial Systems (DoCEIS), Jul 2023, Monte Da Caparica, Portugal. pp.82-94, 10.1007/978-3-031-36007-7_6. hal-05473074

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# Superconducting Saturable Core Reactor as Variable Inductance for Controlling the Power Flow in a Transmission Line

Leonardo Miúdo¹,²✉, João Murta Pina¹, and Nuno Amaro¹

¹ NOVA School of Science and Technology, Center of Technology and Systems (UNINOVA-CTS) and Associated Lab of Intelligent Systems (LASI), NOVA University Lisbon, 2829-516 Lisbon, Portugal

l.miudo@campus.fct.unl.pt

² Instituto Politécnico, Universidade Katyavala Bwila, 1725 Benguela, Angola

**Abstract.** Worldwide electric energy consumption is increasing, partially due to the electrification of sectors such as transportation. Simultaneously, there is a need to implement the ongoing energy transition with renewable energy generation, which results in increased transmission capacity needs. This paper uses steady-state simulations (power flow analysis) in an IEEE standard electrical grid of three buses using a saturable core reactor (SCR) with a high-temperature superconducting (HTS) DC coil, on a 2-core model, to optimize the power flow. The SCR-HTS works as a variable inductance, thus having power flow control capabilities, where superconducting materials are used to substantially decrease losses when compared to conventional conductors. Different scenarios are simulated in MATLAB using a developed Newton-Raphson (NR) algorithm and validated using PSSE. Results indicate that the SCR-HTS can effectively control the power flow in transmission lines and can be used as a solution to further integrate renewable energy sources in the electrical grid and to ensure safe operating conditions in contingency cases.

**Keywords:** Superconducting · Power Flow · Saturable Core Reactor · Transmission Line

## 1 Introduction

Energy and electricity consumption are related to the growth of the world's population, both increasing significantly [1, 2], with electrification expected economic growth so that global electricity demand will grow rapidly in the coming decades [3]. Renewable energy sources have been one of the solutions to meet the increase in consumption globally as well as one of the main keys to the ongoing energy transition. With the insertion of new energy sources in the grid, existing overhead transmission lines need to have a greater transmission capacity and greater control of power flow. The construction of new energy transport infrastructure is a complex process because aspects such as
voltage level, number of circuits, provisions of conductors (line geometries), among others, should be considered [4] and, with the existing fast pace of installation of new generations units, there is the risk of having non-optimal solutions.

Dynamic Line Rating (DLR), Flexible AC Transmission System (FACTS), and the Saturable Core Reactor (SCR) with a High-Temperature Superconducting (HTS) DC coil (as an alternative to FACTS for power flow control) have been widely studied and considered effective means to improve capacity and control in power flow to integrate renewable energy and/or in contingency cases. Regarding the thermal limits of overhead lines, the DLR methods still have some limitations in applications. The physical model (Model IEEE Standard 738 and CIGRE Standard [5]) is of simple implementation and versatile, but the accuracy is relatively low. While other models need a lot of data and versatility is limited [6]. For stability reasons, FACTS, the Distributed Static Series Compensator (DSSC) has relatively low cost and high reliability. However, the control capacity is limited because it can only inject reactive power. Combined devices based on Voltage Source Converter (VSC), Unified Power Flow Controller – UPFC and Distributed Power Flow Controller – DPFC, have better power flow control capability and are therefore the most suitable devices. However, its high cost and complexity become the main limitations of its practical application [7]. SCR emerges as an alternative to FACTS, although SCR involves the use of power electronics devices such as FACTS, the most important difference between both is that the AC and DC circuits of the SCR have no direct electrical connection and are coupled simply through the magnetic field in the core. In SCR, there is not power flows along electronic power components. Consequently, only low-voltage electronic components are necessary for control, while FACTS controllers use electronic power components that are also part of the main power circuit [8]. SCR-HTS DC coil is based on the characteristic of materials to have low resistivity in the superconducting state, using a superconducting material with DC coils in the SCR can obtain large DC magnetomotive force with low losses compared to SCR with conventional conductors.

This paper presents a model and prototype of an SCR-HTS, which is used for power flow control purposes in a line. In order to demonstrate the effectiveness of the solution, a three-bus system is simulated, and results are discussed. The structure of this paper is as follows; Sect. 2 presents the contribution of this article seeing the electricity grid as a cyber-physical space. Section 3 takes an SCR-HTS approach as a variable inductance based on the two-core model first presenting the magnetic equations and later the validation of this same approach. Section 4 displays the network model and its data, impedance, voltage, and active and reactive power. As well as the method implemented for power flow analysis. In Sect. 5, the results, of the power flow analysis in the transmission lines are presented using MATLAB and PSSE. Section 6 contains the conclusions and future work.

## 2 Contribution to Connected Cyber-Physical Spaces

Future electric power grid can be defined as a cyber-physical space (CPS) with two layers; the first layer, physical layer, which is constituted by the electrical energy systems, generators, transmission lines and loads, and a second layer, the cyber layer, where the
control and computing devices and communication lines between them are located [9]. The purpose of the cyber layer is to provide control over the electrical grid and improve its performance as well as increase its reliability. The contribution of this paper is the idea of adding to the electrical grid an SCR-HTS as a cyber control to make the transmission line dynamic. SCR-HTS controls power flow and can help in contingency cases for both the integration of renewable energy sources as well as mitigating the market splitting.

## 3 Superconducting Saturable Core Reactor

In the Publication of the United States Navy [10] a SCR is defined as a device used to control the reactance of an AC coil, through the permeability of the core in which the coil is wound. In other words, a SCR is a ferromagnetic core with windings connected to a DC circuit and a second set of windings connecting an AC circuit, according to the Fig. 1.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Basic schematic of a saturable-core reactor [8]</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Design of the SCR with superconducting coil.</figcaption></figure>

In power systems, a SCR can be used in different applications where the effect of a variable reactance is needed, for example, power flow control [11, 12], fault current limiting [13–15], reactive power compensator continuous (with capacitor in series), harmonics filter and to suppress the voltage flicker in power distribution system [8].
SCR-HTS DC coil is based on the characteristic of materials to have low resistivity in the superconducting state. Using a superconducting material with DC coils in the SCR can obtain large DC magnetomotive force with a small loss compared to SCR with conventional conductors. The SCR-HTS usually consists of a superconducting DC coil, non-superconducting AC coils, and an iron core, Fig. 2.

## 3.1 Two-Core Model of an SCR-HTS

Two-core model approach consists of dividing the three-legged cores into two magnetically decoupled identical core, this is an approach based on the analysis of Saturable-Core Fault Current Limiters (SCFCLs) and was used in the study of SCR in [12] and was the model used in the prototype in [11, 14]. Each magnetic core has a uniform magnetic flux that results from the interaction of the dc and ac flux components.

**Magnetic Equations:** The mathematical equation that describes the flux interactions and inductance characteristics of model starts with Ampère's law, which is the relationship between current and magnetic field strength:

$$ \oint \mathbf{H} \cdot d\mathbf{l} = Ni \quad (1) $$

The integral of a closed contour over the fixed length takes us the expression H as a function of the core length and the magnetomotive force. The relationship between flow intensity and the AC and DC parameters of each core result in the following:

$$ H_{\text{core#1}} = \frac{N_{\text{DC}}I_{\text{DC}} - N_{\text{AC}}i_{\text{AC}}}{l_1} \qquad H_{\text{core#2}} = \frac{N_{\text{DC}}I_{\text{DC}} + N_{\text{AC}}i_{\text{AC}}}{l_2} \quad (2) $$

As the core material has nonlinear magnetic characteristic, then the magnetic field (B) can be expressed as:

$$ B = \mu H \quad (3) $$

Considering the uniform magnetic field perpendicular to surface, the total magnetic flux of each magnetic core is obtained by the following:

$$ \emptyset = \int_A \vec{B} \cdot d\vec{A} = BA \quad (4) $$

The flux linkage in terms of the AC and DC currents is expressed by:

$$ \begin{aligned} \Psi &= N_{AC}\emptyset \rightarrow \Psi = N_{AC}\mu H A \rightarrow \Psi \\ &= \frac{N_{AC}A}{l}(N_{DC}I_{DC} \pm N_{AC}i_{AC})\mu_{(I_{DC}, i_{AC})} \end{aligned} \quad (5) $$

The differential inductance results from the combination of the flux linkage and partial derivative in relation to the AC current as follows:

$$ L_d = \frac{\partial \Psi(t)}{\partial i_{AC}} = N_{AC}A \frac{\partial (\mu_{(I_{DC}, i_{AC})} H_{AC}(t))}{\partial i_{AC}} \quad (6) $$
$$L_{d_{total}} = L_{d\_core#1} + L_{d\_core#2} \qquad (7)$$

$$L_{d\_core#1} = \frac{N_{AC}^2 A}{l} \mu_{d\_core#1}(I_{DC}, i_{AC}) \qquad (8)$$

$$L_{d\_core#2} = \frac{N_{AC}^2 A}{l} \mu_{d\_core#2}(I_{DC}, i_{AC}) \qquad (9)$$

Equations 8 and 9 correspond to the inductances as a function of DC saturation current and AC current.

<figure><img src="image_6.png" /><figcaption>Fig. 3. SCR-HTS with the current source and the LCR meter used to measure inductance [11].</figcaption></figure>

**Validation Two-Core Model:** Two-core model was validated in [11], who uses the prototype of a three-phase SCR-HTS, as seen in Fig. 3. The DC coil is placed in a cryostat of stainless steel and filled with liquid nitrogen to provides a constant temperature of 77 K, necessary for the HTS tape to maintain a superconducting state. A DC source was used to saturate the cores. The Fig. 4 shows the behavior of the AC coil inductances, as a function of the DC current, the reactance as a function of the current is shown in the Fig. 5. As region 2 is very sensitive the DC current variation results in a high regulation of the power flow. The inductances of the AC coils vary from 8.4 to 58.5 mH, which corresponds to reactance variation of 2.6 to 18.3 Ω. For analysis of the power flow in the transmission line, Sect. 4, the resistance of the SCR-HTS is negligible, thus considered as a pure inductive reactance.

## 3.2 Other Models

In addition to the two-core model there are other two models for SCR and that can also be applied in SCR-HTS for power flow control, this is the model derived by finite element analysis [16] and gyrator capacitor model [17–19].
<figure><img src="image_7.png" /><figcaption>Fig. 4. Inductance of the AC coils [11]</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Reactance of the AC coils [11].</figcaption></figure>

# 4 Power Flow in Transmission Line

For analysis of the SCR-HTS as a variable inductance for the control of the power flow, a model of a 3-bus grid, consisting of two sources and a load, will be considered. The following Subsection 4.1, details the grid specifications and methods used for analysis of power flow analysis on the transmission line.

## 4.1 Electrical Grid Model

The analysis of the power flow will be done in the system represented in Fig. 5, consisting of three buses (IEEE – 3 Bus System). The base power was considered 100 MVA and the base voltage is 60 kV. From these, the base current, 1.67 kA, and base impedance, 36 Ω, are deducted.

The model considers the resistance and reactance of the lines corresponding to 0.1 Ω/km and 0.3 Ω/km. The distance between the lines is line 12–30 km, line 13–26 km, and line 13–15 km, thus resulting, respectively, at the following line impedances $Z_{12} = 3.0 + j9.0$ Ω, $Z_{13} = 2.6 + j7.8$ Ω, and $Z_{23} = 1.5 + j4.5$ Ω. The system impedance and admittance parameters are those indicated in Table 1 and the load and generation data are presented in Table 2, all represented in the per unit system. In line 1–3 the
<figure><img src="image_9.png" /><figcaption>Fig. 6. Single-line diagram of the three-buses system.</figcaption></figure>

<table><caption>**Table 1.** Impedance admittance of transmission line.</caption><thead><tr><th>Line</th><th>Impedance (p.u.)</th><th>Line Charging Admittance (p.u)</th></tr></thead><tbody><tr><td>1–2</td><td>0.083 + j0.25</td><td>0</td></tr><tr><td>1–3</td><td>0.072 + j0.29</td><td>0</td></tr><tr><td>2–3</td><td>0.042 + j0.125</td><td>0</td></tr></tbody></table>

impedance value (0.072 + j0.29) Ω is considering the SCR-HTS already inserted in the grid, through the sum of the line reactance and the minimum reactance of the SCR-HTS, 2.6 Ω. This line has a transmission capacity equal to 23 MVA and resistance of SCR-HTS is considered to have a negligible value. The maximum and minimum reactive power generation limits on bus 2 are 35 Mvar and 0, respectively. The analysis of the power flow was made using Newton Raphson (N-R) algorithm in rectangular form in MATLAB, since it allows having a fast convergence thus reducing the simulation time and the number of iterations required in relation to the Gauss and Gauss-Seidel methods.

<table><caption>**Table 2.** Voltage and powers of generation and specified load.</caption><thead><tr><th rowspan="2">Bus</th><th rowspan="2">Voltage (p.u.)</th><th colspan="2">Generator</th><th colspan="2">Load</th></tr><tr><th>MW</th><th>Mvar</th><th>MW</th><th>Mvar</th></tr></thead><tbody><tr><td>1</td><td>1.06 + j0.0</td><td>—</td><td>—</td><td>0</td><td>0</td></tr><tr><td>2</td><td>1.00 + j0.0</td><td>20</td><td>—</td><td>0</td><td>0</td></tr><tr><td>3</td><td>—</td><td>0</td><td>0</td><td>60</td><td>25</td></tr></tbody></table>
Two scenarios are considered for the analysis of the power flow, first, with an inductive reactance in the line of 1–3 of 0.29 Ω, considering the minimum reactance of the SCR-HTS and, second, with an increased inductive reactance to 0.725 Ω, considering the maximum reactance of the SCR-HTS, according to the values of the previous section obtained in the validation of two core model plotted considering region 2. For each case, the active and reactive powers, losses, capacity, and currents in the lines will be presented and analyzed.

# 5 Results and Discussion

In this section the analysis of the results will be done by seaming the two scenarios in each of the subsections.

## 5.1 First Scenario

For the first scenario, NR algorithm implemented in MATLAB converged after four iterations. Obtained power flow results are included in Table 3. As all obtained voltages are in the interval of 0.9 to 1.1 p.u., the system is considered as stable. The reactive power of generator 2 is practically equal to zero. As for the power flow in the transmission lines, the results were calculated and obtained according to Table 4. The powers are represented in the p.u. system, it is worth remembering that the base power is 100 MVA, Table 5 shows the currents that flow in the lines.

<table><caption>Table 3. Analysis results by MATLAB – first scenario.</caption><thead><tr><th>|Bus No.|</th><th>|Bus Type|</th><th>|Voltage|</th><th>|Angle|</th><th>|MW (G)|</th><th>|Mvar (G)|</th><th>|MW (L)|</th><th>|Mvar (L)|</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1.06</td><td>0</td><td>0.41459</td><td>0.30065</td><td>0</td><td>0</td></tr><tr><td>2</td><td>3</td><td>1.0209</td><td>-0.024326</td><td>0.2</td><td>0.0078185</td><td>0</td><td>0</td></tr><tr><td>3</td><td>3</td><td>0.99389</td><td>-0.06189</td><td>0</td><td>0</td><td>0.6</td><td>0.25</td></tr></tbody></table>

In the result of system simulation using PSSE software, Fig. 6, the green arrow corresponds to the active power and the orange arrow corresponds to the reactive power represented in the international system (MW and Mvar), the voltages are in p.u. The power when exiting the bus is positive and when entering the bus is negative.

Considering the capacity of line 1–3 equal to 23 MVA, in first scenario the line is overloaded has an apparent power equal to 32.5 MVA that corresponds to about 40% higher than line capacity. As for the losses, adding up to all the lines a total of 1.5 + j5 was obtained and the line that presented the highest amount of loss is line 1–3, 0.7 + j2.8.
<table><caption>**Table 4.** Power flow in lines – first scenario.</caption><thead><tr><th>Line</th><th>P (p.u.)</th><th>Q (p.u.)</th><th>I (p.u.)</th></tr></thead><tbody><tr><td>1–2</td><td>0.14481</td><td>0.11904</td><td>0.17684</td></tr><tr><td>1–3</td><td>0.26978</td><td>0.18161</td><td>0.30681</td></tr><tr><td>2–3</td><td>0.34221</td><td>0.11122</td><td>0.35247</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Power flow for the first scenario.</figcaption></figure>

<table><caption>**Table 5.** Current in the lines – first scenario.</caption><thead><tr><th></th><th>Line 1–2</th><th>Line 1–3</th><th>Line 2–3</th></tr></thead><tbody><tr><th>I (A)</th><td>295.32</td><td>512.37</td><td>588.62</td></tr></tbody></table>

## 5.2 Second Scenario

For the second scenario, by increasing the inductive reactance of the SCR-HTS from line 1–3 from 0.29 Ω to 0.725 Ω to reduce the power flow in this line, also after four interactions the results are obtained according to Table 6. The bus voltages for second scenario are acceptable and within the limits set for the analysis as well.

The variation of the reactance of the SCR-HTS will imply the variation of the power flow, the results of the quantities prove this variation by reducing the power flow in line 1–3 and increasing in line 1–2, Table 7.

In the case of the power of the line for this second scenario, it was reduced to 21 MVA which corresponds to 91% of the line capacity. As for the losses there was an increase from 1.5 + j5 to 2 + j8.1 and this increase was compensated by the sources on bus 1 (active power) and bus 2 (reactive power). Line 2–3 is the one with the highest power loss, 1 + j3, as can be seen in Fig. 7. As for the currents in the rows, there was a
<table><caption>**Table 6.** Analysis results by MATLAB – second scenario.</caption><thead><tr><th>Bus No.</th><th>Bus Type</th><th>Voltage</th><th>Angle</th><th>MW (G)</th><th>Mvar (G)</th><th>MW (L)</th><th>Mvar (L)</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1.06</td><td>0</td><td>0.42077</td><td>0.33225</td><td>0</td><td>0</td></tr><tr><td>2</td><td>3</td><td>0.99561</td><td>-0.048059</td><td>0.2</td><td>0.023717</td><td>0</td><td>0</td></tr><tr><td>3</td><td>3</td><td>0.9568</td><td>-0.10085</td><td>0</td><td>0</td><td>0.6</td><td>0.25</td></tr></tbody></table>

<table><caption>**Table 7.** Power flow in lines – second scenario.</caption><thead><tr><th>Line</th><th>P (p.u.)</th><th>Q (p.u.)</th><th>I (p.u.)</th></tr></thead><tbody><tr><td>1–2</td><td>0.26576</td><td>0.18965</td><td>0.30801</td></tr><tr><td>1–3</td><td>0.15501</td><td>0.14261</td><td>0.1987</td></tr><tr><td>2–3</td><td>0.45789</td><td>0.16593</td><td>0.48917</td></tr></tbody></table>

<table><caption>**Table 8.** Current in the lines – second scenario.</caption><thead><tr><th></th><th>Line 1–2</th><th>Line 1–3</th><th>Line 2–3</th></tr></thead><tbody><tr><th>I (A)</th><td>514.37</td><td>331.83</td><td>816.91</td></tr></tbody></table>

reduction in line 1–3 from 512 A to about 331 A, which causes it to increase in the other lines, Table 8. The SCR-HTS simulation in this paper was for a 60 kV transmission line. One of the barriers to SCR-HTS development has been implementation at high voltage levels, beyond the limits insulation, cooling system development and high complexity and cost [20]. Still there is some research and prototypes in tests of the application of SCR-HTS as a current limiter in 138 kV and 220 kV [21]. The highest voltage found of the SCR-HTS as a current limiter was 500 kV in a single-phase system [21]. Therefore, SCR-HTS has a great advantage that it can work by combining current limiter and power flow controller (Fig. 8).

Analyzing the two scenarios one can see that the integration of the SCR-HTS limited the power flow in line 1–3, which was overloaded. This results in a possible effect in generation, as in a normal operation the generation in Bus 1 would have to be curtailed (without the SCR-HTS) so that the line thermal capacity is respected. By adding the SCR-HTS, it was possible to eliminate the overload situation, thus ensuring that there is no need for curtailment. Assuming that the generator in bus 1 is renewable generation (e.g. wind), is it then possible to conclude that the SCR-HTS would act as a solution to mitigate the need for curtailment and its negative effects in power systems, which included added operation cost (for compensation of curtailment) and environmental impact costs (resulting in the need to produce energy from controllable sources, e.g. thermal).
<figure><img src="image_11.png" /><figcaption>Fig. 8. Power flow for the second scenario.</figcaption></figure>

The SCR-HTS can have improved control over the impedance of the transmission line, giving flexibility in the operation of the grid, allowing optimal management of the power flow, managing grid congestion, and efficiently transferring power from renewable energy sources to other areas of the grid and thus reducing the impact of market division. In addition, in some cases, it is possible to improve the voltage regulation and stability of the network, which helps in the reliability of the power system.

As for the superconducting part of the SCR-HTS, there has been little research on the dynamic voltage on HTS tapes in power applications [22]. In [23] simulations were performed to optimize the energy efficiency in relation to the dynamic voltage in a 4 mm HTS tape with DC currents from 30 A to 60 A, AC magnetic field magnitudes from 90 mT to 360 mT and frequencies from 50 Hz to 1000 Hz.

# 6 Conclusions and Future Work

The results of the simulations showed that it is possible to control the power flow by varying the inductive reactance of the line using an SCR-HTS through a continuous current. Using a superconducting material with DC coils in the SCR can obtain large DC magnetomotive force with a small loss compared to SCR with conventional conductors. Through this variation of the inductive reactance of the line the SCR-HTS can be inserted into the electrical grid for the control of power flow and current limiter in applications, for example, in contingency cases both for the integration of renewable energy sources as well as mitigating the market splitting. As future work is intended to develop a new model of the SCR-HTS for power flow control based on the characteristic linked magnetic flux (Ψ) and current *I* from [13] and a new model based on the gyrator capacitor. Subsequently, control the power flow with SCR-HTS in a dynamic line approach of the transmission line based on weather conditions, for example, deterministic model IEEE 738 Standard and/or CIGRE Standard.
**Acknowledgments.** This research was funded (in part) by the Portuguese FCT program, Center of Technology and Systems (CTS) UIDB/00066/2020 / UIDB/00066/2020 and gratefully thank the Instituto Nacional de Gestão de Bolsas de Estudo (INAGBE) – Angola for financial support.

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