Open‐phase temporary overvoltage before and after an intermediate subsation accessing a long distance transmission line

Funding information Nature Science Foundation of China, Grant/Award Number: 51907145 Abstract In transmission lines, shunt-reactors are commonly applied to counteract the “Ferranti effect.” However, in highly compensated lines, hazardous overvoltage may occur during unbalanced open-phase conditions. In addition, when a new substation is put into operation, π circuit can be selected by accessing an existing transmission line and the original shunt-reactor configuration is maintained. Severe overvoltage may occur on the shuntreactor side of the newly installed transmission line. In this study, we analysed the effects of compensation, neutral-reactor, and system frequency on open-phase overvoltage. The maximum deviation of system frequency is <±0.5 Hz, therefore, we suggested maximum critical compensation of the shunt-reactor under different proportional coefficients of neutral-reactor. We analysed the impact of line division, when an intermediate substation is accessed, and a risk range in which the substation located may cause severe overvoltage was proposed. Finally, according to an actual line division project, we optimised the original configuration scheme of the shunt and neutral-reactors. We used electromagnetic transient simulation to verify whether severe overvoltage occurred before and after the power station is accessed. The study results may provide guidance for selection of the π circuit accessing point and for optimising the configuration of the shunt and neutralreactors.


INTRODUCTION
SHUNT reactors are commonly applied to long-distance transmission lines to limit power frequency overvoltage and highamplitude operation overvoltage [1]. However, during the dead time of single-pole trip-and-reclose (SPTR) or non-synchronous operation of circuit breakers (CBs), the system may be in unbalanced open-phase conditions. The connected phase and shunt reactors on the disconnected phase form a series resonant circuit through interphase and phase-to-ground capacitances [2,3]. Therefore, severe overvoltage may occur under openphase conditions, which may jeopardise equipment safety. An overvoltage incident that occurred on a 500-V 72% shuntcompensated line in the BC Hydro system led to surge arrester failures at both terminals [4]. In addition, a transformer in the sub-network with delta-connected secondary or tertiary windings can also cause excessive overvoltage [5].
power station is put into operation, the solution of the π circuit accesses a long-distance transmission line can be selected. The compensation of the shunt reactor changes according to the length of the transmission line, and severe overvoltage may occur. Therefore, the necessity and safety of a shunt reactor before a new station is installed are of special interest in engineering. Analysing the risk range of an accessing point causing severe open-phase overvoltage is necessary to realise the optimal shunt reactor configuration on a long-distance transmission line in the design stage and to obtain a guide for selecting the accessing point.
In this study, we analysed the effects of shunt reactor compensation, proportional coefficient of the neutral reactor, and system frequency on open-phase overvoltage. We analyse the impact of line division, when an intermediate substation is accessed and the original shunt compensation is maintained in the system, proposed the risk range of an accessing point causing severe open-phase overvoltage for the shunt reactor configuration at one end or both ends. According to an actual project, we optimised the configuration schemes of shunt and neutral reactors and used electromagnetic transient simulation to verify the open-phase overvoltage of the project. No severe open-phase overvoltage was observed before and after the new power substation accessing the transmission line.

MECHANISM OF OPEN-PHASE OVERVOLTAGE
An extra-high-voltage transmission line can experience uneven phase operations during SPTR operations or in case of failure of a CB pole. The voltage is coupled to the disconnected phase from the connected phase through interphase capacitance. When the shunt reactor capacity is improperly configured, a series resonant circuit is formed, and severe open-phase overvoltage may occur.

Single-phase tripping overvoltage
The source side impedance is much smaller than that of transmission line, when it is connected to two power systems of large capacity. On the other hand, the resistance is about one-tenth of inductance for multi-conductor bundles adopted in highvoltage overhead lines. To simplify the derivation in the theoretical analysis, the source side impedance is not taken into account, and the Overhead line is treated as ideal. The schematic of the system during a single-phase trip-out is presented in Figure 1. C M is the interphase capacitance, C D is the phase-to-ground capacitance, L P is the shunt reactor, and L N is the neutral reactor. Equivalent analysis on the connection modes of the shunt and neutral reactors is presented in Figure 2.
The shunt and neutral reactors can be divided into interphase inductance (L M ) and phase-to-ground inductance (L D ), which can be calculated using Equation (1): Therefore, the equivalent circuit diagram of the circuit in Figure 1 is displayed in Figure 3.
For circuit ① in Figure 3(a), the voltage at point A can be calculated using Equation (2): where, U B and U C are the RMS of B and C phase-to-ground voltages, respectively, X D and X M are phase-to-ground and interphase reactance, respectively: . (3) When the parameters match properly, the denominator in Equation (2) approaches zero, and U A approaches infinity: When X D > 0, it is inductive; when X D < 0, it is capacitive. Similarly, when X M > 0, it is inductive; when X M < 0, it is capacitive.

Two-phase tripping overvoltage
The equivalent circuit diagram of the two-phase trip-out of a transmission line is presented in Figure 3(b). The voltage of phases A and B can be calculated as follows: Similarly, if the denominator in Equation (6) approaches zero, we obtain Equation (7):

Selection of a neutral reactor
Assuming that the shunt reactor compensation capacity of a high-voltage long-distance transmission line is Q LP , we obtain Equation (8): The compensation capacity of a shunt reactor can be divided into phase-to-ground compensation capacity (Q LD ) and interphase compensation capacity (Q LM ). The compensation degree k 1 can be expressed using Equation (9): where C 1 is the positive-sequence capacitance of the transmission line. The relationship between positive-sequence, zero-sequence, interphase and phase-to-ground capacitance can be described using Equation (10): When the interphase capacitance is completely compensated, the interphase impedance is infinite, which is equivalent to an open circuit, and open-phase overvoltage does not occur. Therefore, after determining the shunt reactor compensation degree k 1 , positive-sequence capacitance C 1 , and zero-sequence capacitance C 0 of the transmission line, the neutral reactor when the interphase capacitance is completely compensated can be calculated using Equation (11): However, the actual value and the demand value of the neutral reactor may differ because of design and manufacturing error. If the actual value of the neutral reactor is represented by L N ′ , the deviation from the demand value can be expressed using the proportional coefficient k 2 , as follows: In practical engineering applications, the equivalent interphase inductance L M ′ and phase-to-ground inductance L D ′ are expressed using Equation (13): In addition, according to Equation (11), the demand value of the neutral reactor inductance is related to the system angular frequency ω. When the system frequency fluctuates during the switching process, open-phase overvoltage may occur. It is advisable to consider the design and manufacturing error of shunt reactor and neutral reactor during the calculation of openphase overvoltage. In addition, the influence of frequency variation of the power system on the overvoltage in case of a fault must be considered [11].

Open-phase overvoltage on long-distance transmission lines
When open-phase overvoltage occurs on a transmission line, whether it is a single-phase or two-phase trip-out, the two series parts of the circuit in Figure 3  According to Figure 4, to avoid high-amplitude open-phase overvoltage in case of single-phase and two-phase trip-outs (in this case, the open-phase overvoltage amplitude is < 1.0 p.u.), the following conditions should be satisfied: In actual engineering, generally, under-compensation is the compensation strategy of shunt reactors for high-voltage transmission lines. When k 1 < 100%, 1/X D + 3/X M < 0, which may lead to several scenarios: 1. When X D < 0 and X M < 0, n is located in the OB curve (OB′), and the open-phase overvoltage amplitude is < 1 p.u.; 2. When X D < 0 and X M > 0, n is located in the OC curve (OC′), and the open-phase overvoltage amplitude is < 1 p.u.; 3. When X D > 0 and X M < 0, n is located in the CDE curve (C'E′), and the open-phase overvoltage amplitude is > 1 p.u.; Therefore, when k 2 > 1; that is, the neutral reactor is large and the interphase is over-compensated (which corresponds to situation ②), high-amplitude open-phase overvoltage does not occur on the line. When k 2 < 1; that is, the neutral reactor is small and the interphase is under-compensated, X M < 0. If the compensation degree k 1 is small and X D < 0, which corresponds to situation ①, the phase-to-ground and interphase are both under-compensated, high-amplitude open-phase overvoltage does not occur. If the compensation degree k 1 is large and X D > 0, which corresponds to situation ③, high-amplitude open-phase overvoltage may occur.
Partial conclusions can be obtained from Figure 5, as follows: In addition to the shunt reactor compensation and neutral reactor deviations, the system frequency deviation affects openphase overvoltage. When the system frequency is higher than 50 Hz, the design value of the neutral reactor calculated according to Equation (10) is larger, and open-phase overvoltage does not occur. Open-phase overvoltage occurs only when the system frequency is lower and the design value of the neutral reactor is smaller. Considering that the maximum limit of the frequency deviation under normal operating conditions is ±0.5 Hz [12], the critical value of the shunt reactor compensation degree k 1 is calculated when the open-phase overvoltage amplitude of a 500-kV single-circuit transmission line does not exceed 1.0 p.u. for different neutral reactor proportional coefficients. The results are presented in Figure 6.
According to the aforementioned analysis, for a typical singlecircuit high-voltage transmission line without a neutral reactor, when the compensation degree of shunt reactor k 1 < 67%, or

Open-phase overvoltage after a new power station accessing an existing long-distance transmission line
When a new power station accesses an existing transmission line, the compensation degree of the shunt reactor changes as the length of the line changes. If parameters are configured improperly, severe open-phase overvoltage may occur.

Shunt reactor at both ends of the original transmission line
Firstly, we analysed the case in which the shunt reactor is equally distributed at both ends of the original transmission line. Assuming that the length of the original transmission line is l, the relationship between the lengths of the transmission lines after the new power station is accessed is expressed in terms of the introducing coefficient k 0 as follows: When k 0 decreases from 0.5 to 0, X D first decreases from negative to −∞ and then from +∞ to ωL D /2, X M is positive and gradually decreases, and n′ decreases from negative first and then becomes positive, which may cause open-phase overvoltage. When k 0 increases from 0.5 to 1, X D is still negative and increases gradually, X M increases from positive to +∞ and then becomes negative, and n′ increases from negative to positive. Before the new station is accessed, n > −1/3; thus, n′ > −1/3, and open-phase overvoltage does not occur.
In summary, after a new power station accessing an existing long-distance transmission line, severe open-phase overvoltage may occur only when k 0 < 0.5, that is, in case of a short transmission line. After a new station accessing a transmission line, the phaseto-ground and interphase reactance of the transmission line can be expressed using Equation (18): .
Equation (19) can be derived from Equation (18): Figure 7 presents the relationship curve between the introducing coefficient k 0 and the open-phase overvoltage amplitude when the shunt reactor compensation factor k 1 = 80% and the neutral reactor proportional coefficient k 2 = 1.
For a 500-kV single-circuit transmission line with the parameters displayed in Table 1, when k 0 is in the range of 0.35-0.4, a high-amplitude open-phase overvoltage may occur. The shaded part in Figure 8 indicates risk range of k 0 that the open-phase overvoltage amplitude will exceed 1.0 p.u.

Shunt reactor at one end of the original transmission line
In actual engineering, a shunt reactor may be installed only on one end of a long-distance transmission line, generally on the longer line side after the new station accessing the transmission line.
The neutral reactor of the original transmission line assumed to be fully compensated interphase capacitance. After a new sta- According to the analysis of situation ②, after the new station accessing the transmission line, it can be obtained as follows: where X′ D and X′ M are the phase-to-ground and interphase reactance after the new station accessing the transmission line. Because the shunt reactor is installed only at one end of the line, after the new station accessing the transmission line, the phase-to-ground and interphase compensation capacities, Q LD and Q LM , respectively, remain unchanged. Q CD and Q CM are the phase-to-ground and interphase charging powers before the new station accessing the transmission line. Q′ CD and Q′ CM are the phase-to-ground and interphase charging powers after the new station is accessed, which can be obtained using Equation (21): (21) We obtain Equation (22) by substituting Equation (21) into Equation (20): When n′ = −1/3 and k 0 = k 1 , after the new station accessing the transmission line, the open-phase overvoltage of the transmission line with a shunt reactor is 1.0 p.u. If k 0 is slightly larger than k 1 , then C′ D increases, X′ D < 0 and X′ D also increases, then n′ > −1/3; a severe open-phase overvoltage will not occur. If k 0 is slightly smaller than k 1 , then C′ D decreases, X′ D < 0 and X′ D also decreases, then n′ < −1/3; high-amplitude open-phase overvoltage may occur.
The shaded part in Figure 9 represents the value range of k 0 when the amplitude of open phase overvoltage is > 1.0 p.u. in the context of an original 500-kV single-circuit transmission line with a shunt reactor installed on one end of the line; the compensation degrees are 50%, 60%, 70%, 80%, 90% and 95%.
In summary, risk ranges in which the accessing point may cause open-phase overvoltage amplitude > 1.0 p.u. under different shunt reactor configuration schemes are presented in Table 2.

Case study for a new power station accesses a long-distance transmission line
The length of the 500 kV transmission line was 300 km, which was evenly divided into three sections, and a complete cycle transposition was implemented every 100 km. The source side positive sequence impedance is about 10 Ω, and the zero sequence impedance is about 12 Ω. The positive-sequence and   Table 1. The schematic of the transmission line is displayed in Figure 10.
After the transmission line has been in operation for a specific period, an intermediate substation is planned to be accessed as a π circuit. The schematic of line division is displayed in Figure 11. The lengths of the lines after the power station studied in this paper was accessed are 186 and 114 km.

Analysis and simulation verification of open-phase overvoltage of the original transmission line
According to the results of the aforementioned case study, the neutral reactor value of interphase full compensation under different compensation degrees of shunt reactor can be calculated  Demanded neutral reactor when shunt reactor installed at single end (Ω) ≥0 ≥20 ≥68 ≥104 Demanded neutral reactor when shunt reactor installed at both ends (Ω) ≥0 ≥40 ≥136 ≥208 by substituting the transmission line parameters in Equation (11). According to Figure 5, the demanded proportional coefficient k 2 of neutral reactor can be obtained when the open-phase overvoltage amplitude is < 1.0 p.u., as shown in Table 3. The aforementioned analysis was verified using the electromagnetic transient program EMTP/ATP. The 500 kV overhead line 4×JL/LB1A-500/45 was used, and the transmission tower was a typical cup type tower with a height of 40 m. Considering the example of single-phase and two-phase trip-out when the CB was closed to calculate open-phase overvoltage, the switching action time at both ends of the transmission line was t = 0.2 s, the loss of transmission line is considered in the calculation.
Generally, in engineering, the value of neutral reactor is an integer (a multiple of 50 Ω). The value of neutral reactor adopted in simulation and the results are presented in Table 4.
According to the simulation results, when the neutral reactor is selected according to Table 3, the open-phase overvoltage amplitudes are < 1.0 p.u. When the neutral reactor value is nearly that of interphase complete compensation, the openphase overvoltage amplitude reduces further.

Analysis and simulation verification of open-phase overvoltage after a new power station accessing the transmission line
According to the project plan, the k 0 after a new power station accessing the long-distance transmission line are 0.62 and 0.38, respectively. Analysis of Table 2 reveals that when the shunt reactor is single-ended and the compensation is 70%, or when the reactor is double-ended and the compensation is 80%, the amplitude of open-phase overvoltage after the new power station accessing the transmission line may be > 1.0 p.u. In addition, when the reactor is single-ended and the degree of compensation is 60%, or when the reactor is double-ended and the degree of compensation is 70%, k 0 is approximately in the risk range. Considering the change in line parameters after the new power station accessing the transmission line and the error after assuming the integer value of the neutral reactor, highamplitude open-phase overvoltage may also occur in these two shunt reactor configuration schemes.
The accuracy of the aforementioned analysis was verified using simulation calculations with the same input parameters as Chapter A. Table 5 shows the results of the open-phase overvoltage amplitude of the transmission line under different shunt configuration schemes.
The results indicate that in case of double-ended and the compensation degree is 80% or 70%, single-ended and the compensation degree is 70%, the open-phase overvoltage is >1.0 p.u.; in case of single-ended and the compensation degree is 60%, the overvoltage is less than but approximately equal to 1.0 p.u. The simulation results agree well with those of the previous analysis.
To limit power frequency overvoltage, the compensation degree of the shunt reactor should not be too low. However, considering the errors of line parameters, neutral reactor and shunt reactor design parameters and actual parameters, the shunt reactor compensation degree should not be too high. Therefore, considering the current operation and the access of the long-term power station, we recommend that a single-ended configuration be used with a shunt reactor compensation of 80% and a neutral reactor of neutral reactor 500 Ω.

CONCLUSION
In this paper, the effects of the compensation degree of the shunt reactor, neutral reactor and system frequency on openphase overvoltage were analysed. Parameters of a typical 500 kV single-circuit transmission line were considered, and a risk range was determined in which the amplitude of open-phase overvoltage after the new power station accessing the transmission line was > 1.0 p.u. The conclusions of this paper can guide the optimal configuration of the shunt reactor of the original line. In addition, the paper can provide guidance for the selection of the accessing point of the solution of a π circuit when a new power station is put into operation. The main conclusions are as follows:

APPENDIX Appendix A. Derivation of Equation (1)
Equivalent analysis on the connection modes of the shunt and neutral reactors is presented in Figure 12. For circuit in Figure 12(a), the voltage on the nodes can be calculated using Equations (23) to (25): For circuit in Figure 12 The voltage between phase A and phase B in Figure 12 The line voltage in Figure 12 Appendix B. Derivation of Figure 3 Taking the equivalent model in Figure 2 into Figure 1, the schematic of the system during a single-phase trip-out can be equivalent to Figure 13(a). As the concern is the voltage of open phase A, it is mainly related to the voltage of phases B and C, which is fixed and determined by the system. Therefore, the connection between B and C phases can be ignored, the simplified circuit is shown in Figure 13(b). Figure 13(b) is further simplified to Figure 3.