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This paper deals with a new concept for calculating DC harmonic voltages and currents of line- commutated HVDC systems. In contrast to the conventional method, this method is useful for BTB (Back-To-Back) HVDC systems without smoothing reactors or PTP (Point-To-Point) with very short transmission line. This method proposes a new direction for HVDC system design and analysis. The proposed method is applied to a 50 Hz/60 Hz BTB test system and a synchronized BTB test system. After simulation and verification, the new results are introduced.

This conventional line-commutated HVDC converter with cable or overhead lines acts in the AC side as current sources for characteristic currents independently. That is, an inverter is considered as only an ideal current source or voltage source in the viewpoint of a rectifier and inverter is on the contrary. So, in the cases of the harmonic calculation on the DC side and the design of smoothing reactors, considering the other side is meaningless. However, with respect to HVDC without a transmission line (cable or overhead line) such as a BTB system, because the rectifier and inverter in a BTB HVDC system are strongly coupled each other, the influence of the second converter cannot be disregarded. This paper proposes extended HVDC equations considering the coupling effects between rectifiers and inverters. The proposed methods are more suitable than a separated current source model in case of the harmonic current/voltage and minimum DC current calculations.

The definition of currents and voltages used for the calculation are shown in

where,

V_{d}_{1h}, V_{d}_{2h} = DC harmonic voltages of rectifier and inverter in harmonic order (h),

V_{d}_{10}, V_{d}_{20} = No-load dc voltages of rectifier and inverter.

α_{1}, α_{2} = firing angles of rectifier/inverter, μ_{1}, μ_{2} = overlap angles of rectifier/inverter,

h = 12, 24, 36, 48 ××× etc. (12-pulse, integer).

The resulting harmonic voltages in the DC circuit will be calculated according to the following equation [

The resulting source voltage comprising both rectifier and inverter harmonics depends on the phase angle j_{0} between both AC systems. In general, this value is based on both system configurations and load flow conditions and will therefore not be a constant value. As an average usually a value of j_{0} = 90˚ will be assumed for system and design studies.

The DC current consists of two parts as indicated in

where Id_{total} = mean value of the DC current obtained assuming hypothetical infinite inductance on the DC side of the circuit, Id_{ripple} = mean value of the DC ripple, Id_{offset} = part of the Id current not related to the DC current ripple.

The existence of DC current discontinuities depends on the operating conditions, converter configurations, and system frequencies. Normally, Id_{total} at minimum power is selected to give a positive value of Id_{offset}.

The classical expression for calculating the maximum Id_{ripple} due to the influence of a single 12-pulse converter is found in [

where,_{ripple} : maximum operating DC current ripple, E_{LL }_{max pk} : maximum valve winding voltage, peak, 6-pulse converter,_{cr} and L_{ci}: 6-pulse commutating reactances of the rectifier and inverter, respectively.

The expression above is calculated by assuming no overlap. For this particular analysis, where Id is at its minimum level, the overlap is normally very small and the approximation is quite acceptable. Since for BTB converters the influence of the second converter cannot be disregarded, Equation (5) can be extended to:

where, α_{r} = Firing angle of the rectifier, α_{i}_{ }= Firing angle of the inverter.

On the inverter side, α_{i} is defined as:

where, μ_{i} = Overlap angle of the inverter, γ_{i} = Gamma angle of the inverter.

Since the overlap angle is assumed to be zero,

The equations presented until now describe the mean value of the DC current, but are not sufficient to explain the nature of the current ripple in HVDC schemes in the time and frequency domains. The time domain contribution due to a single 6-pulse converter is calculated in [

Equation (9) and Equation (10) represent the individual contributions to id_{ripple}(t) considering a single pulse from each converter side. Switching functions are required to completely describe id_{ripple}(t) for any time period.

Before starting, we assume the following: firstly, for this analysis the AC system voltage assumed at the commutation busbar is the positive sequence fundamental frequency voltage. This is a reasonable assumption for the dominant harmonics since the harmonic filter design will ensure that the harmonic content at the commutation busbar is minimal.

Secondly, assuming that α + μ is limited to 45˚ during steady-state operation, the equations above also represent a good approximation of the maximum harmonic ripple along the full power range. This is based on the well-known behavior of the DC side ideal harmonic voltage source that varies as a function of the firing angle and overlap as indicated in [

As shown in

_{ripple}(t) and related DC quantities based on the time domain switching

functions. This example is for a BTB system connecting a 50 Hz system to a 60 Hz system. The plots in shown in

For calculating idr(t) and idi(t) in _{HV}, is assumed zero. idr(t) and idi(t) are defined in Equation (9) and Equation (10), respectively, and vt(t) is defined in Equation (11). This test example assumes that the rectifier is at 50 Hz, and the inverter is at 60 Hz.

This difference originates from the fact that the rectifier and inverter in the BTB HVDC system are strongly coupled to each other. Therefore, because DC harmonics calculation and the smoothing reactor design are incorrect in the classical method [

The DC voltage shown in

where, vr(t)_{p.g} = ideal 12-pulse DC voltage for the rectifier (pole to ground), vi(t)_{g.p} = ideal 12-pulse DC voltage for the inverter (ground to pole), and θ_{r} and θ_{i} are the AC system phase voltage angles of the equivalent independent sources at the rectifier and the inverter, respectively.

_{ripple}(t), for the 50 Hz/60 Hz case shown in _{ripple}(t), which appears on the DC side as shown in

and the overlapped waveforms are shown as mentioned.

In a BTB HVDC system, except for the overlap effect of frequencies, the difference between the AC system phase voltage angles, (θ_{r} ‒ θ_{i}), can impact the maximum and minimum amplitudes of id_{ripple}(t). The (θ_{r} ‒ θ_{i}) values that will generate the extreme values of id_{ripple}(t) are not so easy to predict, because the AC voltage angle is always varied according to AC load conditions.

From Equation (11), for converters with identical frequencies (synchronous connection, 60 Hz/60 Hz BTB), the AC voltage system angles will impact the harmonics on id_{ripple}(t) as follows:

-For (θ_{r} ‒ θ_{i}) = 0, if we assume perfect symmetry between the two sides, the harmonic spectra generated by the rectifier and inverter will have the same phase angle and will add in phase.

-For (θ_{r} ‒ θ_{i}) = 7.5˚, if we assume perfect symmetry between the two sides, the harmonic spectrum generated by the rectifier will be phase shifted by 7.5˚. For the 24th harmonic current, the period is 360˚/24 = 15˚. So if the rectifier and inverter contributions are phase shifted by 7.5˚ at the 24th harmonic, they will cancel out when added.

For (θ_{r} ‒ θ_{i}) = 15˚, if we assume perfect symmetry between the two sides, the harmonic spectrum generated by the rectifier will be phase shifted by 15˚. For the 12th harmonic current, the period is 360˚/12 = 30˚. So if the rectifier and inverter contributions are phase shifted by 15˚ at the 12th harmonic, they will cancel out when added.

Similar to

From these results, we can verify that the optimal smoothing reactor to reduce the DC harmonic current can be calculated considering the different frequency conditions and the difference in angles, (θ_{r} ‒ θ_{i}).

Also, in

synchronous interconnection (60 Hz).

Because the rectifier and inverter in a BTB HVDC system are strongly coupled to each other, a new coupled analysis model is needed. The conclusions of analysis by the method proposed in this paper are as follows.

· The total DC current ripple, id_{ripple}(t), which appeared on the DC side, overlapped the characteristic harmonic due to the 50 Hz AC source with the characteristic harmonic due to the 60 Hz AC source.

· The characteristic harmonic of the lower AC frequency side is always greater whether the lower frequency side is a rectifier or not, and is not related to the values of the smoothing reactor.

· For (θ_{r} ‒ θ_{i}) = 0, id_{ripple}(t) is maximum.

· For (θ_{r} ‒ θ_{i}) = 7.5˚, the 24th harmonic will be minimal or zero.

· For (θ_{r} ‒ θ_{i}) = 15˚, the 12th harmonic will be minimal or zero.

I thank Dr. Mike Li of ALSTOM for his support and advice for this paper.