Performance Investigation and Optimization of a Novel Hybrid Saturated-Core Fault-Current Limiter Considering the Leakage Effect

Abstract

To reduce the requirement of DC-biasing capacity and improve the biasing ability of a permanent magnet (PM), a novel hybrid saturated-core fault-current limiter (HSCFCL) is proposed in this paper. Compared with traditional saturated-core fault-current limiter (SCFCL), the HSCFCL has the advantages of small size, low DC-biasing capacity, a high biasing ability of the PM and excellent limiting performance. Firstly, the principle and the magnetic circuit model of the HSCFCL are introduced. Then, the improvement of DC-biasing capacity with a PM is analyzed. In addition, the influence of the leakage-flux effect on the biasing ability of the PM is presented in detail, and a small-section optimal structure is proposed to improve the biasing ability of the PM. Finally, to validate the principle and performance of the HSCFCL, various electromagnetic simulations, optimization studies and experiments are carried out. The simulation and experimental results demonstrate the effectiveness of the proposed method.

1. Introduction

Since renewable energy, energy demand and the scale of power grids continue to grow, excessive short-circuit current has been a serious problem for power systems [1,2,3,4]. Traditional current-limiting measures such as an air-core reactor and network splitting have some disadvantages [5]. Fault-current limiter devices (FCL) are becoming a promising technology to limit fault current. There are several types of FCLs [6,7,8]. The saturated-core fault-current limiter (SCFCL) has attracted worldwide attention from researchers and companies and has the advantages of fault self-detection, fast action and a high withstanding voltage [9,10,11].

The SCFCL utilizes the non-linear permeability of ferromagnetic materials to provide a low impedance under normal conditions and a large limiting impedance under fault conditions in order to limit the fault current [12]. Under normal conditions, the DC magnetic-motive-force (MMF) produced by DC coils drives the cores into saturation, and the impedance of SCFCL is low. During a fault event, the large fault current drops the cores out of saturation, resulting in a large increase of impedance for limiting the fault current. However, there are two major problems limiting the development of traditional SCFCLs.

The first problem is the large DC-biasing capacity of DC coils under normal conditions. Superconducting coils and permanent magnets (PMs) can be used to solve the problem. However, the superconducting materials and cooling system are expensive and energy consuming [13,14,15]. In addition, the biasing capacity of PMs is inadequate [16]. To reduce the requirement of the DC-biasing capacity of traditional SCFCLs, a hybrid DC coil/PM design in an open-core SCFCL is discussed [17]. However, the reduction of the fault-clipping performance needs to be improved. Modelled on an open-core type, a hybrid 4-limb close-core SCFCL based on PMs is discussed [18]. But the structure is complex and large in size. Moreover, there is a lack of detail in comparative studies about the impact of PMs on DC-biasing capacity and the fault-clipping performance of PMs.

Additionally, due to the PM and the deep saturation of cores, the leakage-flux effect is obvious, and has a significant influence on the performance of the SCFCL and the biasing ability of the PM. An analytical model of a PM fault-current limiter device (PMFCL) considering the leakage-flux effect is established in [19]. A small-section structure of the PMFCL is proposed to improve the biasing ability of the PM in [20]. However, there has been a lack of attention to the biasing ability of the PM that considers the leakage effect and the optimization of a small-section structure in a hybrid DC coil/PM structure.

The second problem for existing SCFCLs is insufficient limiting performance. Due to the coupling effect between the AC and DC windings, even though the cores are driven out of saturation, the limiting impedance of the SCFCL is still not large, resulting in insufficient limiting performance [21]. A bridge-type SCFCL with limiting inductor is proposed to improve limiting performance [22].

This paper proposed a novel hybrid saturated-core fault-current limiter (HSCFCL), which has the advantages of small size, low DC-biasing capacity, a high biasing ability of the PM, and excellent limiting performance. The hybrid DC coil/PM structure is proposed to reduce the DC-biasing capacity. A small-section optimal structure is proposed to improve the biasing ability of the PM. A limiting inductor in series with the DC-biasing circuit is proposed to improve the limiting performance. Firstly, the fundamental principle and the magnetic circuit model of HSFCCL are introduced in Section 2. Then, Section 3 presents the biasing ability analysis of the PM considering the leakage effect in detail. Finally, various simulations, optimization studies and experiments are performed in Section 4. The simulation and experimental results verify the effectiveness of the proposed structure. In Section 5, conclusions are summarized.

2. Fundamental Principle and Magnetic Circuit Model of the HSCFCL

To describe the principle conveniently, the main symbols and definitions in this paper are firstly given, as shown in

Table 1. Symbols and definitions.

Symbol	Definition	Symbol	Definition
φm1	Flux of cores I	Xnom	Normal impedance of HSCFCL
φm2	Flux in cores II	ZFCL	Inserted impedance of HSCFCL
φm3	Flux in cores III	L0	Limiting inductance
re1	Reluctance of core I	MMFd	DC MMF produced by DC coils
re2	Reluctance of core II	MMFPM	DC MMF produced by PMs
ru	Reluctance of core III	∆MMFPM	Improvement of DC MMF
μu	Unsaturated permeability of iron core	Id	DC current
μsr	Saturated permeability of iron core	Se	Cross-sectional area of core I
Lu	Unsaturated inductance of a winding	le	Length of core I
Ls	Saturated inductance of a winding	Sb	Cross-sectional area of small section
N	Number of winding turns	lb	Length of small section

.

2.1. Basic Configuration of the HSCFCL

Figure 1 shows a basic configuration of the HSCFCL, which has three cores, AC coils, DC coils, PMs, a limiting inductor and a biasing DC source. The hybrid biasing structure of the DC coils and PMs is used to drop the cores into saturation, resulting in a reduction of the DC-biasing capacity. The MMF produced by the PMs and DC coils are in same direction. In order to limit both the positive and negative half cycles of a fault-current effectively, the DC MMFs of cores I and II are in opposite directions. The limiting inductor L₀ which is in series with the biasing DC source is used to enhance the limiting performance. The small-section parts in cores I and II are used to improve the biasing ability of the PMs. The cross-sectional area of the small-section part is reduced to S_b (which is smaller than that of the main core S_e); the length of the small-section part is l_b.

2.2. Magnetic Circuit Modeling

According to the configuration of the HSCFCL, magnetic flux loops of cores and PMs have been established, as shown in Figure 2a. The DC flux set up by the DC coils φ_dc and the DC flux set up by the PMs φ_PM mainly flow through cores I and II, and the AC flux mainly flows through core III. Figure 2b shows an equivalent magnetic circuit model of the HSCFCL. To simplify the analysis, assume that the magnetization curve of the iron core is a double-line model. The saturated permeability of the iron core is equal to that of air μ₀, and the unsaturated permeability of the iron core is μ_u. φ_m1, φ_m2 and φ_m3 represent the fluxes in cores I, II and III, respectively. r_e1, r_e2 and r_u represent the reluctances of core I, II and III, respectively.

Assume that the AC is in the positive half cycle. Using the loop current method:

$$\{\begin{array}{l}-2H_c{l}_m+2r_m{\varphi }_{m1}+H_{e1}{l}_e-N{I}_{ac}-N{I}_d+r_{u3}({\varphi }_{m1}-{\varphi }_{m2})=0\\ -2H_c{l}_m+2r_m{\varphi }_{m2}-H_{e2}{l}_e+N{I}_{ac}-N{I}_d+r_{u3}({\varphi }_{m2}-{\varphi }_{m1})=0\\ {\varphi }_{m3}={\varphi }_{m1}-{\varphi }_{m2}\end{array}$$

(1)

where H_e1 is the magnetic field strength of core I; H_e2 is the magnetic field strength of core II; l_e is the length of the magnetic path of core I or core II; N is the number of winding turns; I_ac is the alternating current through the HSCFCL; and I_d is the DC biasing current.

Under normal conditions, cores I and II are dropped into deep saturation by DC MMFs produced by the DC coils and PM. I_ac1 = I_ac2 = I_ac, r_e1 = r_e2 = r_s = l_e/A_zμ_sr, H_e1 = H_e2 = H_e. Hence, the fluxes in core I, II, and III can be calculated by solving (1):

$$\{\begin{array}{c}{\varphi }_{m1}=\frac{2H_c{l}_m+N{I}_d}{2r_{\mathrm{m}}}+\frac{N{I}_{ac}-H_e{l}_e}{2(r_m+r_{u3})}\\ {\varphi }_{m2}=\frac{2H_c{l}_m+N{I}_d}{2r_m}-\frac{N{I}_{ac}-H_e{l}_e}{2(r_m+r_{u3})}\\ {\varphi }_{m3}={\varphi }_{m1}-{\varphi }_{m2}=\frac{N{I}_{ac}-H_e{l}_e}{(r_m+r_{u3})}\end{array}$$

(2)

It can be seen that the DC MMF produced by the PMs and DC coils mainly flow through cores I and II. Core III mainly transports the AC MMF, as shown in Figure 3a. Moreover, although the DC MMF is enhanced with the PMs, the reluctance of the DC magnetic loop also increases. Hence, the improvement of DC-biasing capacity by PMs needs to be further studied.

Since φ_m1, φ_m2, H_e and I_ac are functions of time t, the saturated inductance of a winding can be expressed as:

$$L_s=N\frac{d{\varphi }_{m1}}{dt}/\frac{d(I_{ac}/2)}{dt}=\frac{2N^2}{2(r_m+r_{u3})+r_s}$$

(3)

The unsaturated inductance of a winding can also be expressed as:

$$L_u==\frac{2N^2}{2(r_m+r_{u3})+r_u}$$

(4)

Hence, the normal impedance of HSCFCL is:

$$X_{nom}=\omega \cdot L_s=\frac{2\cdot \omega \cdot N^2}{2(r_m+r_{u3})+r_s}$$

(5)

Since the permeability of the cores in saturation and the PM are approximately equal to that of air, the normal impedance of the HSCFCL under normal conditions is very low.

Under fault conditions, assume that the fault current is in the positive half cycle. Then, core I is saturated, and core II is unsaturated. The fault current flows through the windings of core I and the limiting inductor L₀. Hence, the inserted impedance of the HSCFCL under fault conditions Z_FCL can be expressed as:

$$Z_{FCL}=\omega \ast (2L_s+L_0)=\omega \ast (\frac{4N^2}{2(r_m+r_{u3})+r_s}+L_0)$$

(6)

2.3. Improvement of DC-Biasing Capacity with PMs

Compared with traditional SCFCLs with DC coils, the hybrid structure of the DC coils and PMs can reduce the DC current of the DC coil, resulting in a reduction of the DC-biasing capacity and losses.

To drive cores into saturation under normal conditions, using the loop current method:

$$N{I}_d+2H_c{l}_m-2r_m{\varphi }_s-H_s{l}_e=N{I}_{cm}$$

(7)

where I_cm is the critical normal current.

Assume the DC MMF produced by the DC coils MMF_d = NI_d, the DC MMF produced by PMs MMF_PM = 2H_cl_m. Since H_c = B_r/μ_m, r_m = l_m/(μ_mS_m), the DC MMF produced by the DC coils can be expressed as:

$$MM{F}_d=N{I}_{cp}+H_s{l}_e+MM{F}_{PM}(1-\frac{B_s{S}_e}{B_r{S}_m})$$

(8)

Without PMs, the DC MMF of the traditional SCFCL which is needed to drive the cores into saturation is:

$$MM{F}_d^{\prime }=N{I}_{cp}+H_s{l}_e$$

(9)

Thus, the improvement of the DC MMF with PMs can be calculated as:

$$\Delta MM{F}_d=MM{F}_d^{\prime }-MM{F}_d=MM{F}_{PM}(1-\frac{B_s{S}_e}{B_r{S}_m})$$

(10)

It can be seen that, if B_sS_e/(B_rS_m) < 1, the improvement is positive. Hence, the HSCFCL can reduce the DC-biasing capacity when appropriate parameters of cores and PM materials are designed. The materials of the core and PM in this paper are Grade Silicon steel 30Q140 (H_s = 10 kA/m, B_s = 2.0 T and NdFeB N50 (H_c = 836 kA/m, B_r = 1.45 T), respectively. Thus, the cross-sectional area of the PM and cores should be satisfied as:

$$\frac{S_m}{S_e}>\frac{B_s}{B_r}=1.379$$

(11)

In addition, to ensure excellent limiting performance under fault conditions, the limiting inductance L₀ should be less than kL_u, k is 0.01~0.1. Hence:

$$\frac{X_{FCL}-2X_{nom}}{\omega }\le k{L}_u=\frac{k{N}^2}{2r_m+2r_u}$$

(12)

Since r_u is much less than r_m, r_u can be ignored. Hence, to ensure the limiting performance, MMF_PM has a maximum value:

$$MM{F}_{PM}\le \frac{\omega k{N}^2{B}_r{S}_m}{20(X_{FCL}-X_{nom})}=MM{F}_{P\max }$$

(13)

By increasing the MMF_PM, the improvement of the DC MMF will increase. When MMF_PM = MMF_Pmax, the improvement of the DC MMF is maximum, and the length of PM can be calculated as:

$$l_m=\frac{\omega k{N}^2{B}_r{S}_m}{2H_c(X_{FCL}-X_{nom})}$$

(14)

3. Biasing Ability Analysis of the PM Considering the Leakage Effect

Under normal conditions, cores I and II are driven into deep saturation. The leakage flux will make the flux density distribution of the cores uneven. Moreover, due to the PMs, the leakage effect of the HSCFCL is more obvious and should be considered. Figure 3 shows the leakage-flux density distribution of the air region near the HSCFCL without small sections.

It can be seen that the leakage effect of the HSCFCL is significant. The leakage-flux density near the PM is largest with about 0.15 T. In addition, the leakage-flux density near cores I and II is smaller, but the leakage-flux density distribution is uneven, resulting in uneven flux density distribution of the cores. The flux density in the central part of the cores is smaller than that in the terminal part of the cores close to the PMs.

The significant leakage effect will severely weaken the biasing ability of the PMs and the improvement of DC-biasing capacity. Moreover, the uneven flux density distribution of cores I and II will reduce the saturation degree of the cores, leading to a reduction in the biasing ability of the PMs and an improvement of DC-biasing capacity. Hence, the influence on the biasing ability of the PMs considering the leakage effect should be analyzed and improved.

To simplify the analysis, the HSCFCL is equivalent to a 2D model, ignoring the edge effect. According to the principle of magnetic field division, the air region above the HSCFCL is divided in six types of cylindrical flux tubes with different cross-sectional areas, which are referred to as the arch type A₁, similar circular type A₂, semi-lunar type A₃, semi-circular type A₄, semi-lunar type A₅, and semi-circular type A₆, as shown in Figure 4.

3.1. Arch-Type Flux Tube A₁

The arch-type flux tube represents the leakage flux emitted from line HO₁ of the PM to line IO₁. According to the principle of magnetic-field division, the permeance of the arch-type flux tube A₁ can be calculated as:

$$G_{A1}=\frac{{\mu }_0{W}_d{\theta }_1}{12}$$

(15)

where W_d is the width of PM; and θ₁ is the angle of arc HI.

3.2. Similar Circular-Type Flux Tube A₂

The similar circular type flux tube A₂ represents the leakage flux emitted from line JH of the PM to line IK. According to the infinitesimal method, the permeance of the similar circular type flux tube A₂ can be calculated as:

$$G_{A2}={\displaystyle \underset{r_1}{\overset{r_2}{\int }}\frac{{\mu }_0{W}_d}{r\pi }}dr$$

(16)

where r₁ and r₂ are the radius of arcs HI and JK, respectively.

3.3. Semi-Lunar Type Flux Tube A₃

The semi-lunar type flux tube A₃ represents the leakage flux emitted from point J to point K. The permeance of the semi-lunar type flux tube A₃ can be calculated as:

$$G_{A3}=\frac{{\mu }_0{W}_d(\pi -{\theta }_2)}{12}$$

(17)

where θ₂ is the angle of arc JK.

3.4. Semi-Circular Type Flux Tube A₄

The semi-circular type flux tube A₄ represents the leakage flux emitted from side LJ of the iron core to side KM. Point M is the center of core III. The permeance of the semi-circular type flux tube A₄ can be calculated as:

$$G_{A4}=\frac{{\mu }_0{W}_d}{\pi }\ln \frac{\mathrm{LM}}{\mathrm{JP}}$$

(18)

where LM and JP are the length of lines LM and JP.

3.5. Semi-Lunar Type Flux Tube A₅

The semi-lunar type flux tube A₅ represents the leakage flux emitted from point L of PM1 to point P of PM2. The permeance of the semi-lunar type flux tube A₅ can be calculated as:

$$G_{A5}=\frac{{\mu }_0{W}_d(\pi -{\theta }_3)}{12}$$

(19)

where θ₃ is the angle of arc LP.

3.6. Semi-Circular Type Flux Tube A₆

The semi-circular type flux tube A₆ represents the leakage flux emitted from surface side NL of the iron core to surface side PQ. The permeance of the semi-circular type flux tube A₆ can be calculated as:

$$G_{A6}=\frac{{\mu }_0{W}_d}{\pi }\ln \frac{\mathrm{NQ}}{\mathrm{LP}}$$

(20)

where NQ and LP are the length of lines NQ and LP.

In addition, the air region below the PMs can be divided into three magnetic tubes: A₇, A₈ and A₉, as shown in Figure 4c. The permeance of the three magnetic tubes can be expressed as:

$$G_{A7}=\frac{{\mu }_0{W}_d{\theta }_4}{12}$$

(21)

$$G_{A8}={\displaystyle \underset{r_4}{\overset{r_5}{\int }}\frac{{\mu }_0{W}_d}{r\pi }}dr$$

(22)

$$G_{A9}=\frac{{\mu }_0{W}_d(\pi -{\theta }_5)}{12}$$

(23)

where θ₄ and θ₅ are the angles of arcs O₁′O₃′ and J′O′, respectively; and r₄ and r₅ are the radius of arcs O₁′O₃′ and J′O′, respectively.

It can be seen that the permeances of tube A₁, A₂, A₃ and A₄ are parallel, the permeances of tube A₅ and A₆ are parallel, and the permeances of tube A₇, A₈ and A₉ are parallel. Assuming G₁ = G_A1 + G_A2 + G_A3 + G_A4, G₂ = G_A5 + G_A6, G₃ = G_A7 + G_A8 + G_A9, G_m is the permeance of PM. Hence, the equivalent permeance model of the PMs considering the leakage effect is shown in Figure 5.

According to Thevenin equivalent principle, Thevenin equivalent MMF and reluctance of PM can be expressed as:

$$F_m^{\prime }=H_c{l}_m\cdot \frac{G_m}{G_m+G_1+2G_2+2G_3}$$

(24)

$$R_m^{\prime }=\frac{1}{G_m+G_1+2G_2+2G_3}$$

(25)

Hence, due the leakage effect, the equivalent MMF of PM will decrease, resulting in a reduction of the biasing ability of the PM. The length of the PM that is needed should be longer than that of Equation (14) when considering the leakage-flux effect.

Hence, to improve the biasing ability of the PM and reduce the length of the PMs, a small section part is proposed. Due to the cross-sectional area being smaller than that of main core, this can make the flux-density distribution of the cores more even. The cross-sectional area and length of the small-section part should be studied.

4. Results

To validate the performance and improvement of the DC-biasing capacity of a HSCFCL in a power grid, a single-phase 10 kV/1 kA HSCFCL finite element analysis (FEA) model was established. Various FEA simulations using the non-linear field-circuit coupling method and optimization studies of the HSCFCL were performed. In the simulation setup, the material characteristics of the cores were the actual non-linear model. An ideal voltage source was used as the power source, and an ideal switch was used as the circuit breaker. The parameters of the HSCFCL model are presented in

Table 2. Parameters of the HSCFCL model.

Parameter	Value
Cross-sectional area of cores I and II/m2	0.2025
Cross-sectional area of cores III/m2	0.261
Cross-sectional area of small section/m2	0.189
Length of cores le/m	2.2
Length of small section lb/m	1.8
Cross-sectional area of PM/m2	0.405
Length of PM/m	0.15
Number of winding turns N	37
Inductance of limiting inductor L0/mH	1.93
Steel cores	Hs = 10 kA/m, Bs = 2.0 T
Permanent magnets	Hc = 836 kA/m, Br = 1.45 T

. To validate the influence of the PMs, the SCFCL without PMs whose other parameters are the same was performed for comparison. The parameters of the PM and the small-section part have been optimized.

4.1. FEA Simulation and Optimization

Figure 6 shows the DC current and normal voltage of the HSCFCL and SCFCL without PMs under normal conditions. It can be seen that, compared with SCFCL without PMs, the DC current can be reduced from 2.1 kA to 1.0 kA. Hence, the HSCFCL has a 52.3% reduction in DC-biasing capacity. Additionally, because of the reluctance of the PMs, the normal voltage can also be reduced from 179.0 V to 156.0 V with a 12.8% reduction.

The current of the HSCFCL is shown in Figure 7. It shows that the fault current is limited from 30 kA to 11.0 kA with the HSCFCL. A HSCFCL with L₀ = 0 mH represents the traditional hybrid FCL. It can be seen that, compared with traditional hybrid FCLs, the proposed HSCFCL has better limiting performance. In addition, compared with the SCFCL without PMs, the limiting performance is weakened by 10%. However, the HSCFCL also has good limiting performance, meeting the limiting requirement of the power grid.

Figure 8 shows the currents of the coil on core II under different limiting inductances. It can be seen that the limiting performance of the HSCFCL can be improved when the limiting inductance increases. However, due to the reluctance of the PMs, the unsaturated inductance of the winding is not so large compared with the limiting inductance. Hence, with an increase in the limiting inductance, the current of the coil on the unsaturated core will also increase, resulting in a weakening of limiting performance. Compared with the SCFCL with the same limiting inductance L₀ = 1.93 mH, the current of the HSCFCL coil is larger. Hence, Figure 7 and Figure 8 demonstrate that the reluctance of the PMs will have some impact on the limiting performance of the HSCFCL.

Figure 9 shows the flux density distribution of the HSCFCL under different conditions. Under normal conditions, cores I and II are both saturated, the flux density of cores I and II is about 2.05 T, and the flux density of the PM is 1.12 T. Under fault conditions, cores I and II are alternately driven out of saturation, respectively.

To validate the influence of a small-section structure on the saturation degree of the cores and biasing ability of the PMs, various comparative studies with different parameters of small sections are analyzed. The DC current is 1.0 kA.

Figure 10 shows the flux-density distribution of core I in different cross-sectional areas of the small section. It can be seen that, without the small section, due to the leakage effect the flux-density distribution of core I is uneven, the minimum B is 1.984 T in the center, and core I is not saturated. The use of the PM or the DC-biasing capacity should be increased to drop the cores into saturation. The length of the PM should be increased to 0.22 m for driving the cores into saturation. With the small section, saturation is deep and the biasing ability of the PMs is enhanced. When S_b = 0.1935 m² and S_b = 0.1912 m², the minimum B can be enhanced to 1.9969 T and 1.9984 T, but core I is still unsaturated. When S_b = 0.189 m², the minimum B is 2.0003 T, and core I is saturated. However, when S_b is reduced to 0.1867 m², the flux density of the core center increases, but the flux density of the core end decreases to 1.9923 T, resulting in a weakening of the saturation degree. Hence, with the reduction of cross-sectional area of the small section, the saturation degree of the cores does not always increase, and is relatively optimal in S_b = 0.189 m².

Figure 11 shows the flux-density distribution of core I in different lengths of the small section. When l_b = 1.9 m and l_b = 2.0 m, the flux density can be improved to 1.9982 T and 1.996 T, but core I is still unsaturated. When l_b = 1.8 m, core I is saturated with 2.0003 T. However, when the length is reduced to 1.7 m, the flux density near the junction of the small section and core I is reduced to 1.9868 T. Hence, with the reduction in length of the small section, the saturation degree of cores does not always increase, and is relatively optimal at l_b = 1.8 m.

This demonstrates that the small-section optimal structure can improve the saturation degree of cores, resulting in a reduction in the use of PMs and DC-biasing capacity. Moreover, the cross-sectional area and length of the small section have a significant influence on the improvement. The optimal parameter of the small section is S_b = 0.189 m², l_b = 1.8 m. The length of the PM for traditional hybrid SCFCLs without the small section should be increased to 0.22 m to drive the cores into saturation. Hence, compared with traditional hybrid SCFCLs without a small section, the length of the PM can be reduced from 220 mm to 150 mm. The biasing ability of the PM can be improved by 31.8% with the optimal small-section structure.

In addition, the energy consumption of the DC-biasing circuit under normal conditions has an important influence on the performance and cost of the HSCFCL. Reducing the energy consumption of the DC-biasing circuit under normal conditions can also improve the efficiency of the HSCFCL. As discussed previously, the DC-biasing current of SCFCLs without PMs is 2.0 kA, and the DC-biasing current of the HSCFCL can be reduced to 1.0 kA. Moreover, when the length of the PM is 150 mm, the DC-biasing current of traditional hybrid SCFCLs will increase to 1.2 kA. Hence, assuming the lifetime of a FCL is 30 years (263,000 h), and each coil has a resistance of 0.05 $\mathrm{\Omega }$, the energy consumption can be calculated as:

• SCFCLs without PMs: W_loss1 = $4\times {2000}^2\times 0.02\times 263=84.16$ GWh
• Hybrid SCFCLs without small section: W_loss2 = $4\times {1200}^2\times 0.02\times 263=30.30$ GWh
• HSCFCL with small section: W_loss3 = $4\times {1000}^2\times 0.02\times 263=21.04$ GWh

Hence, compared with SCFCLs without PMs, the proposed HSCFCL has a 75% reduction of energy consumption under normal conditions. Compared with traditional hybrid SCFCLs without a small section, the proposed HSCFCL has a 30.5% reduction of the energy consumption, resulting in an improvement in the efficiency of the HSCFCL under normal conditions.

The parameters of the PMs have an important impact on the improvement of the DC-biasing capacity. Moreover, since the reluctance of the PM will reduce the unsaturated inductance of the core, the parameters of PMs also have a significant impact on the limiting performance of the HSCFCL. Hence, to validate the influence of the PM on the performance of the HSCFCL, various comparative studies with different parameters of PMs are performed. The DC current is 1.0 kA, the parameter of the small section is S_b = 0.189 m², and l_b = 1.8 m.

Figure 12 shows the flux-density distribution of core I in different lengths of PMs. When the length of the PM is 0.13 m, 0.15 m and 0.17 m, the minimum flux density is 1.9977 T, 2.0003 T and 2.0036 T, respectively. Thus, as the length of the PM increases, the saturation degree also increases.

Figure 13 shows the limiting performance in different lengths of PM. When the lengths of the PM are 0.13 m, 0.15 m and 0.17 m, the peak fault current can be limited to 10.77 kA, 11.00 kA and 11.34 kA, respectively. If the length of the PM increases, unsaturated inductance will decrease, and the length of the PM has a significant influence on the limiting performance. Hence, increasing the length of the PM can improve the saturation degree and reduce the DC-biasing capacity but will increase the cost and weaken the limiting performance.

Figure 14 and Figure 15 show the flux-density distribution of cores and the limiting performance in different cross-sectional areas of the PM. It can be seen that as the cross-sectional area of PM increases, the saturation degree also increases effectively. When the cross-sectional area of the PM increases from 0.382 m² to 0.427 m², the minimum flux density also increases from 1.9959 T to 2.0029 T. Moreover, the cross-sectional area of the PM has little impact on the limiting performance.

Additionally, compared with Case 1: S_m = 0.405 m², l_m = 0.15 m, Case 2: S_m = 0.405 m², l_m = 0.17 m can reduce the DC capacity of 0.1 kA, but will increase the usage of the PM of 0.0081 m³, and weaken the limiting performance of 0.34 kA. However, Case 3: S_m = 0.427 m², l_m = 0.15 m can also reduce the DC capacity of 0.1 kA, but will just increase usage of the PM of 0.0031 m², and has little impact on the limiting performance. Hence, compared with an increase in the length of the PM, an increase in the cross-sectional area of the PM has better performance and economy.

4.2. Experimental Study

To validate the performance and optimization studies of the HSCFCL proposed by this paper, a 220 V/10 A single-phase HSCFCL laboratory prototype was tested, as shown in Figure 16a. The parameters of the experimental prototype are shown in

Table 3. Parameters of the 220 V/10 A HSCFCL laboratory prototype.

Parameter	Value
Cross-sectional area of cores I and II/m2	0.0025
Cross-sectional area of cores III/m2	0.00325
Cross-sectional area of small section/m2	0.00244
Length of cores le/m	0.27
Length of small section lb/m	0.19
Cross-sectional area of PM/m2	0.005
Length of PM/m	0.01
DC-biasing current/A	10
Inductance of limiting inductor L0/mH	8
Steel cores	Hs = 10 kA/m, Bs = 2.0 T
Permanent magnets	Hc = 836 kA/m, Br = 1.45 T

. The electrical circuit of the 220/10 A prototype test platform is shown in Figure 16b. The power supply was provided by a transformer. The prospective peak-fault current without FCL was 500 A. The DC-biasing current was adjusted using a 100 A bridge rectifier. The exciting source was connected to the rectifier through an exciting transformer to ensure electrical insulation. A 400 A breaker was used to simulate a fault across the load. The currents were measured using 400 A AC/DC closed-loop Hall effect current sensors.

Figure 17 shows the experimental limiting performance of the HSCFCL. Under normal conditions, the current is 10 A. When a fault occurs, the HSCFCL can limit the fault current effectively. Due to the influence of the aperiodic component of the fault current, the fault current has a transient decay process. The first peak fault current with the HSCFCL can be limited to about 120 A. The fault current is limited to approximately 70 A when it reaches a steady state. Additionally, the DC-biasing current of the HSCFCL under normal conditions is 10 A. However, the DC-biasing current is 20 A without PMs. Hence, the DC-biasing capacity can be reduced by 50%.

Figure 18 shows the fault current and DC-biasing current when the fault current reaches the steady state. It can be seen that the frequency of the DC-biasing current is 100 HZ. Due to the reluctance of the PM, the fault current will not only flow through the DC-biasing circuit, but also the windings in the unsaturated core. The DC-biasing current is a little smaller than the fault current. Hence, the experimental results validate the effectiveness of the proposed structure.

5. Conclusions

This paper presents a novel hybrid saturated-core fault-current limiter (HSCFCL), which has the advantages of small size, low DC-biasing capacity, high biasing ability of the PM, and excellent limiting performance. Compared with traditional SCFCLs, the DC-biasing capacity can be reduced by 50% and the biasing ability of the PM can be improved by 31.8% with optimal small-section structure.

The hybrid DC coil/PM structure is proposed for reducing the DC-biasing capacity. As the MMF_PM increases, the improvement of the DC MMF will increase. Considering the limiting performance, the improvement of the DC MMF has a maximum value. In addition, according to the biasing ability of the PM when considering the leakage effect, the leakage effect will weaken the biasing ability of the PM, and the flux-density distribution of the cores is also uneven, resulting in an increase in the usage of the PM and DC-biasing capacity.

The small-section optimal structure can improve the biasing ability of the PM. The cross-sectional area and length of the small section have a significant influence on the level of improvement. Additionally, the cross-sectional area and length of the PM also have an important influence on the improvement of DC-biasing capacity and limiting performance. Compared with an increase in the length of the PM, an increase in the cross-sectional area of the PM can yield better performance and economy.