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Generalized S-transform is a time-frequency analysis method which has higher resolution than S-transform. It can precisely extract the time-amplitude characteristics of different frequency components in the signal. In this paper, a novel protection method for VSC-HVDC (Voltage source converter based high voltage DC) based on Generalized S-transform is proposed. Firstly, extracting frequency component of fault current by Generalized S-transform and using mutation point of high frequency to determine the fault time. Secondly, using the zero-frequency component of fault current to eliminate disturbances. Finally, the polarity of sudden change currents in the two terminals is employed to discriminate the internal and external faults. Simulations in PSCAD/EMTDC and MATLAB show that the proposed method can distinguish faults accurately and effectively.

More and more VSC-HVDC projects are using cables as transmission lines. Due to the small damping of VSC-HVDC System, DC cable fault will cause serious over-current in a short time, which will lead to serious faults on transmission cables and the converter. In addition, VSC-HVDC system has some problems, such as fault identification, self-healing and so on [

The S transform is a time-frequency analysis method, which derived from continuous wavelet transform and short-time Fourier transform. The S transform of a signal x ( t ) is expressed as:

S ( f , t ) = ∫ − ∞ ∞ | f | 2 π e − f 2 ( τ − t ) 2 2 x ( t ) e − j 2 π f t d t = ∫ − ∞ ∞ ω ( f , τ − t ) x ( t ) e − j 2 π f t d t (1)

where: ω ( f , τ − t ) = | f | 2 π e − f 2 ( τ − t ) 2 2 ω ( f , τ − t ) is Gaussian window function

with a time-shift factor τ and a frequency f. The S transform of a discrete signal x [ m T ] is expressed as:

S [ n N T , m T ] = ∑ k = 0 N − 1 X [ k + n N T ] e − 2 π 2 k 2 n 2 e j 2 π k m N . (2)

S [ 0 , m T ] = 1 N ∑ m = 0 N − 1 x [ m T ] . (3)

where m = 0, 1, 2, …, N − 1, N is the number of sampling points, and T is the sampling time interval. Based on the S transform, the Generalized S-transform can be obtained and described as follow:

S ( f , τ ) = ∫ − ∞ ∞ | f α | | β | 2 π e − f 2 α ( τ − t ) 2 2 β 2 x ( t ) e − j 2 π f t d t = ∫ − ∞ ∞ ω ( f , τ − t ) x ( t ) e − j 2 π f t d t (4)

where: ω ( f , τ − t ) = | f α | | β | 2 π e − f 2 α ( τ − t ) 2 2 β 2 . In Equation (4), α and β are regulator,

positive numbers. By adjusting parameters of α or β, superior time-frequency characteristic of the signal is obtained.

The cross section of bipolar cable is shown in

Reference [

and 2, respectively. Taking terminal 1 as an example, u_{p}_{1c}, u_{p}_{1s} and u_{p}_{1a} represents the DC voltage of conductor layer, sheath layer and armor layer of the positive cable at terminal 1 respectively. u_{n}_{1c}, u_{n}_{1s}, u_{n}_{1a} are the DC voltage of conductor layer, sheath layer and armor layer of the negative cable at terminal 1 respectively. The same applies to terminal 2.

Define R, L, G and C as the matrices of the resistance, inductance, conductance and capacitance per meter of the DC cable respectively, and the wave equations with mutual inductance can be represented as follows:

{ ∂ U 1 ∂ x = − R I 1 − L ∂ I 1 ∂ t ∂ I 1 ∂ x = − G U 1 − C ∂ U 1 ∂ t (5)

U 1 = [ u p 1 c , u n 1 c , u p 1 s , u n 1 s , u p 1 a , u n 1 a ] , I 1 = [ i p 1 c , i n 1 c , i p 1 s , i n 1 s , i p 1 a , i n 1 a ] . The key to realizing the decoupling is to transform R, L, G and C to diagonal matrices. According to this principle, the voltage phase mode transform matrix can be derived and represented as follows:

P = k ⊗ [ I I I I I 0 I 0 0 ] (6)

where 0 and I are the 2-dimensional zero matrix and unit matrix respectively, while k is the decoupling transform matrix expressed as

k = [ 1 1 1 − 1 ] (7)

Introduce Q = P^{−T} as the current phase mode transform matrix, and Equation (3) can be transformed to

{ ∂ U 1 m ∂ x = − P − 1 R Q I 1 m − P − 1 L Q ∂ I 1 m ∂ t ∂ I 1 m ∂ x = − Q − 1 G P U 1 m − Q − 1 C P ∂ U 1 m ∂ t (8)

where I_{1m} is expressed as:

I 1 m = Q − 1 I 1 = [ i p 1 c + i n 1 c + i p 1 s + i n 1 s + i p 1 a + i n 1 a i p 1 c − i n 1 c + i p 1 s − i n 1 s + i p 1 a − i n 1 a i p 1 c + i n 1 c + i p 1 s + i n 1 s i p 1 c − i n 1 c + i p 1 s − i n 1 s i p 1 c + i n 1 c i p 1 c − i n 1 c ] (9)

According to [

Taking a two-terminal VSC-HVDC as an example, we build its system simulation model and some typical faults are carried out in PSCAD/EMTDC. The fault model is shown in _{1}, p_{2}, n_{1}, n_{2} are relay protection devices. Fault F_{1} is at the middle point of the positive cable. F_{2} is the bipolar short-circuit fault. F_{3} and F_{4} are fault in the AC system, respectively. In this paper, F_{1} and F_{2} are internal faults, F_{3} and F_{4} are external faults.

The parameters of rectifier station and inverter station are shown in

The 6-th mode current is employed to establish criteria. The Generalized S-transform result matrix of 6-th mode current is marked as G S 6 . When a fault occurs, there are abundant frequency components in the DC cable. The summation of the magnitude of high frequency (5 kHz - 10 kHz) at the time kT can be calculated as:

S u m = ∑ n 1 ≤ n ≤ n 2 | G S 6 [ k + 1 , n + 1 ] | (10)

where n 1 f 0 = 5 kHz , n 2 f 0 = 10 kHz .The criteria that the first sudden change time of S_{um} is the time of fault based on numerous experiments. When the VSC-HVDC operates normally, S_{um} is 0. So the criteria of when the fault occurs can be built as follows: when S_{um} > S_{umh}, it indicates that there is a fault. S_{umh} is the threshold which is obtained in this case: the fault resistance is set to 500 (Ω) and during the remote fault. In this paper, S_{umh} is 0.5. The disturbance caused by noise can also cause high frequency in the cable. To estimate the disturbance, the zero frequency of the G S 6 is employed to establish criteria, which can be obtained as follows:

G S 0 = | G S 6 [ k + 1 , 1 ] | (11)

The Δ G S 0 is the difference of 6-th mode result after Generalized S-transform when VSC-HVDC operates normally and when there is a fault or disturbance.

Station Parameter | Rectifier station | Inverter station |
---|---|---|

AC voltage (kV) | 420 | 420 |

Rated frequency (Hz) | 60 | 50 |

Rated capacity (MVA) | 100 | 100 |

Based on mounts of experiments, the criteria can be built as following: when | Δ G S 0 | > h , it indicates there is a fault, otherwise, it is disturbance. Where h is threshold which should be smaller than | Δ G S ′ 0 | . | Δ G S ′ 0 | is the difference of 6-th mode current after Generalized S-transform when VSC-HVDC operates normally and when there is a remote fault with the largest resistance. In this paper, k is 0.8.

It is necessary to identify the external fault from internal fault because the external fault can also cause the sudden change of sum. Take the monopole to earth fault as an example. The simplified diagram of the system after fault F_{1} is shown in _{f} is the fault resistance and ∆u_{f} is the additional voltage source at the fault point. The voltage between the fault point and the ground is ∆u. Z_{p}_{12} is the equivalent impedance of cable_{12}. Z_{s}_{1} and Z_{s}_{2} are the equivalent impedance of AC sides, respectively. Measured by the protection devices p_{1} and p_{2}, the fault current component is named as ∆i_{p}_{1} and ∆i_{p}_{2}. The reference directions of the DC currents are represented by the arrows displayed in the figure. Seen from _{p}_{1}, ∆i_{p}_{2} can be expressed in Equation (12). From Equation (12), the polarities ∆i_{p}_{1} and ∆i_{p}_{2} are opposite.

{ Δ u = − Δ i p 1 × ( Z p 12 2 + Z s 1 ) Δ u = Δ i p 2 × ( Z p 12 2 + Z s 2 ) (12)

The simplified diagram of the system after an external fault F_{3} at the rectifier side is shown in _{s}_{1f} is the equivalent impedance from the fault point to DC-link capacitor. The relation of ∆u, ∆i_{p}_{1}, ∆i_{p}_{2} can be expressed in Equation (13). Obviously, ∆i_{p}_{1} and ∆i_{p}_{2} have the same polarity.

{ Δ u = Δ i p 1 × ( Z s 1 f + Z p 12 + Z s 2 ) Δ u = Δ i p 2 × ( Z s 1 f + Z p 12 + Z s 2 ) (13)

The flowchart of the protection method is expressed as follows: (

Internal faults are composed of monopole to earth fault and bipolar short-circuit fault. Numerous internal faults are simulated to verify the reliability of the proposed protection method, which are shown in _{1} as an example, which is shown in _{1}, IN_{1}, IP_{2}, IN_{2} are the currents after fault measured at p_{1}, n_{1}, p_{2}, n_{2}, respectively. The 6-th mode current I_{1} and I_{2} are calculated from them. S_{um}_{1} and S_{um}_{2} are shown in (c). at 0.5s, S_{um}_{1} and S_{um}_{2} are both reach the sudden change point and the peak values are larger than S_{umh}. ∆GS_{01} and ∆GS_{02} are both larger than h, which are shown in (d). As shown in (e), the polarities of ∆i_{1} and ∆i_{2} are opposite, which indicates that there is an internal fault in the DC cable.

The simulation results of external faults are shown in

Simulation results for internal faults | ||||
---|---|---|---|---|

Fault location | F1 | F1 | F1 | F2 |

Resistance (Ω) | 0 | 200 | 500 | 0 |

S_{um}_{1} (p.u.) | 20.6 | 2.4 | 0.53 | 21.3 |

S_{um}_{2} (p.u.) | 3.3 | 0.61 | 0.12 | 23.7 |

ΔGS_{01} (p.u.) | 59.4 | 4.3 | 2.4 | 60.2 |

ΔGS_{02} (p.u.) | 1.2 | 1.7 | 1.8 | 59.3 |

Δi_{1} (p.u.) | 300.1 | 30.2 | 7.6 | 280.3 |

Δi_{2} (p.u.) | −80.2 | −10.2 | −2.1 | −291.5 |

Simulation results for external faults | ||||||
---|---|---|---|---|---|---|

Fault location | F3 | F3 | F3 | F4 | F4 | F4 |

Fault type | Single-phase fault | Two-phase fault | Three-phase fault | Single-phase fault | Two-phase fault | Three-phase fault |

S_{um}_{1} (p.u.) | 13.2 | 21.3 | 23.8 | 6.3 | 7.8 | 7.9 |

S_{um}_{2} (p.u.) | 5.1 | 7.6 | 7.8 | 15.7 | 23.2 | 24.1 |

ΔGS_{01} (p.u.) | 13.7 | 15.1 | 15.3 | 6.9 | 7.2 | 7.3 |

ΔGS_{02} (p.u.) | 2.8 | 3.4 | 3.7 | 18.2 | 19.3 | 19.4 |

Δi_{1} (p.u.) | −250.7 | −300.6 | −301.9 | 130.3 | 160.3 | 148.2 |

Δi_{2} (p.u.) | −140.4 | −160.5 | −150.3 | 250.7 | 300.8 | 320.7 |

This paper proposed a novel protection method for VSC-HVDC based on Generalized S-transform. Firstly, using Generalized S-transform to extract frequency component of fault current and employing mutation point of high frequency to determine the fault time. Secondly, eliminating disturbances using the zero-frequency component of fault current. Finally, the polarity of sudden change currents in the two terminals is employed to discriminate the internal and external faults. Simulations in PSCAD/EMTDC and MATLAB show the accuracy and effectiveness of the proposed method.

The authors declare no conflicts of interest regarding the publication of this paper.

Man, W.S., Bei, X.M. and Zhang, Z.Y. (2021) A Protection Method of VSC-HVDC Cables Based on Generalized S-Transform. Energy and Power Engineering, 13, 1-10. https://doi.org/10.4236/epe.2020.134B001