### Characteristic analysis of transient energy

#### Simplified equivalent circuit of HVDC transmission system

To simply analyze, the converter may be approximate equivalent to a series of voltage source and impedance, this equivalent circuit does not produce an obvious influence on the fault analysis. Figure 1 shows the equivalent circuit of single-pole HVDC system, where *Z*
_{
eq
} denotes the equivalent impedance of the converter station; *Z*
_{
L
} is the impedance of DC line; *u*
_{
m+
} and *i*
_{
m+
} are the voltage and current at terminal M, respectively; *u*
_{
n+
} and *i*
_{
n+
} are the voltage and current at terminal N, respectively; *u*
_{1} and *u*
_{2} are the equivalent voltage sources at two converters. Here, supposing the positive direction of current is from DC bus to DC line, as shown in Fig. 1. The main fault types for bipolar DC line include the single pole to earth faults and pole to pole fault, where the single pole to earth faults contain the positive pole to earth fault and the negative pole to earth fault. The analyses of internal and external faults for HVDC line are described as below.

#### Transient energy characteristics for pole-earth fault

If a pole-earth fault occurs on the positive pole line, Fig. 2 shows an equivalent fault superimposed circuit.

In Fig. 2, *L* denotes the length of DC line; F is the fault point; *Z*
_{
x
} is the line impedance from F to terminal M, and *Z*
_{(l
-x)} is the line impedance from F to terminal N; *R*
_{F} is the fault resistance; *u*
_{F} represents the fault superimposed voltage source; Δ*u*
_{m+
} and Δ*i*
_{m+
} are the fault components of voltage and current at terminal M, respectively; Δ*u*
_{n+} and Δ*i*
_{n+
} are the fault components of voltage and current at terminal N.

According to Fig. 2, the following equations can be obtained:

$$ \varDelta {u}_{\mathrm{m}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{m}+} $$

(1)

$$ \varDelta {u}_{\mathrm{n}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{n}+} $$

(2)

From (1) and (2) it can be seen that the polarities of Δ*u*
_{m+
} and Δ*i*
_{m+
} are opposite, so are the polarities of Δ*u*
_{n+} and Δ*i*
_{n+
}. Therefore, the variations of transient power on both sides of the line *ΔP*
_{m +} = *Δu*
_{m +}
*Δi*
_{m +} and *ΔP*
_{n +} = *Δu*
_{n +}
*Δi*
_{n +} are all negative, further their integral values shown in (3) and (4) are also negative.

$$ {S}_{\mathrm{m}+}={\displaystyle {\int}_0^{\tau}\varDelta {P}_{\mathrm{m}+}(t) dt} $$

(3)

$$ {S}_{\mathrm{n}+}={\displaystyle {\int}_0^{\tau}\varDelta {P}_{\mathrm{n}+}}(t) d t $$

(4)

where, *S*
_{m+} and *S*
_{n+} denote the transient energies measured at terminals M and N of the positive pole line, respectively; *τ* is the integral time. Here, the reason that introduces the transient power integral is to decrease the influence from disturbance and improve the reliability of protection.

Similarly, if there is a pole-earth fault occurring on the negative pole line, the transient energies *S*
_{m-} and *S*
_{n-} detected at terminals M and N of the negative pole line are also negative.

#### Transient energy characteristics for pole-pole fault

If a bipolar line fault occurs on the HVDC lines, the fault superimposed circuit is shown in Fig. 3, where F_{1} and F_{2} are the fault points on the positive pole line and negative pole line, respectively. Δ*u*
_{m-} and Δ*i*
_{m-} are the fault components of voltage and current at terminal M of the negative pole line; Δ*u*
_{n-} and Δ*i*
_{n-} are the fault components of voltage and current at terminal N.

According to Fig. 3, the following equations can be obtained:

$$ \left\{\begin{array}{c}\hfill \varDelta {u}_{\mathrm{m}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{m}+}\hfill \\ {}\hfill \varDelta {u}_{\mathrm{n}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{n}+}\hfill \\ {}\hfill \varDelta {u}_{\mathrm{m}-}=-{Z}_{eq}\varDelta {i}_{\mathrm{m}-}\hfill \\ {}\hfill \varDelta {u}_{\mathrm{n}-}=-{Z}_{eq}\varDelta {i}_{\mathrm{n}-}\hfill \end{array}\right. $$

(5)

From (5) it can be seen that the variations of transient power on each side of bipolar line are all negative and, after the integral calculation, the transient energies are also negative.

Based on the above analyses, the following conclusions can be drawn: for an internal DC line fault, whatever it is a single pole to earth fault or a pole to pole fault, the power variations on both sides of the faulted line are always negative, as well as the transient energies.

#### Transient energy characteristics for external fault

For HVDC line, external faults include two types, one type occurs on the DC side, and the other type occurs on the AC side. Actually, the influence generated by an AC side external fault on HVDC line protection is similar to that generated by a DC side external fault, and the transient analyses for the external faults are discussed as follows.

### Faults outside the positive pole line

If a fault occurs on the converter valve or DC bus of the rectifier side, the fault superimposed circuit is shown in Fig. 4a, where *Z*
_{
eq1} and *Z*
_{
eq2} are the equivalent impedances from the fault point to both sides of the rectifier station. For a fault occurring on the AC side at the rectifier station, the fault superimposed circuit is shown in Fig. 4b.

From Fig. 4, Δ*u*
_{m+
} and Δ*i*
_{m+
} can be expressed as:

$$ \varDelta {u}_{\mathrm{m}+}=\left({Z}_{eq}+{Z}_L\right)\varDelta {i}_{\mathrm{m}+} $$

(6)

$$ \varDelta {u}_{\mathrm{n}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{n}+} $$

(7)

In (6) and (7), the polarities of Δ*u*
_{m+
} and Δ*i*
_{m+
} at the rectifier side are the same, while those of Δ*u*
_{n+
} and Δ*i*
_{n+
} at the inverter side are opposite. Therefore, the transient power variation Δ*P*
_{m+
} is positive, but Δ*P*
_{n+
} is negative and, thus the transient energy *S*
_{m+
} is positive while *S*
_{n+
} is negative.

When a fault occurs on the converter valve or DC bus of the inverter side, the corresponding fault superimposed circuit is shown in Fig. 5a and b illustrates the fault superimposed circuit for a fault occurring on the AC side.

As shown in Fig. 5:

$$ \varDelta {u}_{\mathrm{m}+}=-{Z}_{eq}\varDelta {i}_{\mathrm{m}+} $$

(8)

$$ \varDelta {u}_{\mathrm{n}+}=\left({Z}_{eq}+{Z}_L\right)\varDelta {i}_{\mathrm{n}+} $$

(9)

From (8) and (9) it can be seen that the polarities of Δ*u*
_{m+
} and Δ*i*
_{m+
} at the rectifier side are opposite, while those of Δ*u*
_{n+
} and Δ*i*
_{n+
} at the inverter side are the same. Therefore, the transient power variation Δ*P*
_{m+
} is negative, and Δ*P*
_{n+
} is positive, so the corresponding transient energy *S*
_{m+
} is negative while *S*
_{n+
} is positive.

### Faults outside the negative pole line

Similarly, the same conclusions as the aforementioned cases can be drawn: for the faults outside the negative pole line, no matter a fault occurs on the rectifier station or on the inverter station, the polarities of the transient energies detected on both sides of the negative pole line are always opposite.

### Protection criterion for HVDC line

According to the analyses in Section 2.1, the following conclusions can be obtained: in case of an internal fault occurring on the HVDC line, the transient energies detected on both the rectifier side and the inverter side are all negative; but for a fault occurring outside the HVDC line, the transient energy on one side is positive, and the transient energy on the other side is negative.

Therefore, using the characteristic difference of transient energy after the fault, the fault direction can be determined. If the transient energy is negative, the fault direction is positive; if the transient energy is positive, the fault direction is negative. As a result, if the fault directions on both sides of HVDC line are all positive, an internal fault will be discriminated. However, If the fault direction on one side is positive, and that on the opposite side is negative, an external fault will be identified.

It has been specified that the substitution of transient energy for fault component power is to improve the reliability of protection.

To implement conveniently, the discretized transient energy equation of (3) and (4) can be expressed:

$$ {S}_{\eta p}=\varDelta t{\displaystyle \sum_1^j\varDelta {u}_{\eta p}(k)\varDelta {i}_{\eta p}(k)} $$

(10)

where, Δ*u*
_{
ηp
}(*k*) and Δ*i*
_{
ηp
}(*k*) are the fault components of voltage and current at the *k*th sampling point, which can be extracted by filtering load component; *j* is the number of the sampling point used for integral calculation; Δ*t* is the sampling interval; *η* denotes the rectifier side M or the inverter side N; the variable *p* denotes + or -, which means the positive pole line or the negative pole line. In actual calculation process, Δ*t* may be omitted.

If a single pole to earth fault occurs, the transient signals on the faulted line can be coupled to the normal line, which possibly affects the fault identification of the normal line. Moreover, lightning stroke or normal operation can also generate disturbance signals. To avoid these influences, a fixed energy threshold *S*
_{
set
} is necessary, which is defined as follows:

$$ {S}_{set}= j{k}_{rel}{k}_i{k}_u $$

(11)

where *k*
_{
rel
} is the reliable coefficient; *k*
_{
i
} and *k*
_{
u
} are the proportional coefficients of rated current and rated voltage, respectively. Note that the rated current and the rated voltage are all expressed as per unit value here, so that *S*
_{
set
} can be applied to different HVDC system. In addition, for a long HVDC line, the probability of lightning stroke is high. Therefore, lightning stroke line without causing fault must be correctly identified by the protection. The sampling frequency of the proposed method is relatively low (10 kHz) and integral calculation is used in the algorithm, which can filter high frequency components to some extent. Moreover, the amplitude of lightning current for direct stroke line is low [20, 21], and generally the transient energy caused by lightning disturbance will not exceed the fixed threshold.

For convenient analysis, a rule is defined here. The logic value of a positive directional fault is 1, and that of the negative directional fault is equal to −1, and 0 means there is no fault. Therefore, the transient energy direction *Dir*[*S*
_{
ηp
}] is expressed as follows:

$$ D i r\left[{S}_{\eta p}\right]=\left\{\begin{array}{l}-1\kern1em {S}_{\eta p}\kern0.5em \ge {S}_{set}\\ {}\kern0.5em 0\kern1.5em \left|{S}_{\eta p}\right.\kern0.5em \left|<{S}_{set}\right.\\ {}\kern0.5em 1\kern1.5em {S}_{\eta p}\kern0.5em \le -{S}_{set}\end{array}\right. $$

(12)

The logic identifying fault is described as below: if the logic values of fault direction detected on both sides of HVDC line are equal to 1, an internal fault will be determined; if the logic value of fault direction on one side is 1, while that on the other side is −1, an external fault will be discriminated; otherwise, it is normal.