The definition of currents and voltages used for the calculation are shown in Figure 1. In this case, the characteristic 12-pulse DC harmonics voltages for an HVDC converter station will be calculated. The magnitude will be calculated according to the following equations [1] :

where,

V_d1h, V_d2h = DC harmonic voltages of rectifier and inverter in harmonic order (h),

V_d10, V_d20 = No-load dc voltages of rectifier and inverter.

α₁, α₂ = firing angles of rectifier/inverter, μ₁, μ₂ = 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 [2] .

The resulting source voltage comprising both rectifier and inverter harmonics depends on the phase angle j₀ 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 j₀ = 90˚ will be assumed for system and design studies.

The DC current consists of two parts as indicated in Figure 2 and in Equation (4).

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 [3] .

where,: interval where the expression is defined, Id_ripple : maximum operating DC current ripple, E_{LL max pk} : maximum valve winding voltage, peak, 6-pulse converter,: minimum value of the total inductance of the DC circuit, L_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, , and Equation (6) can be rewritten as:

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 [3] . For BTB 12-pulse converters, the ripple can be split into two contributions: one due to the rectifier and the other due to the inverter. Each individual contribution can be expressed in the range using the following expressions for the rectifier and inverter, respectively:

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 [3] .

As shown in Figure 3, the 12th harmonic of the ideal voltage DC source is calculated as a function of the overlap variation considering different alpha values. From Figure 3, the maximum harmonic occurs for the case with minimum overlap and maximum firing angle.

Figure 4 shows an example of the id_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 Figure 4 are not to scale and are presented for facilitating the understanding of the relationship between different electrical quantities. Figure 4 presents a diagram where the electrical quantities of Figure 5 are identified in the circuit.

For calculating idr(t) and idi(t) in Figure 5, the DC total resistance, R_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.

Figure 4 shows the simulation results of the proposed method. In Figure 4, DC voltage, A, and DC current ripple, B, show differences between the asynchronous BTB system (50 Hz/60 Hz) and synchronous BTB system (60 Hz/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 [2] , the coupling influences between the rectifier and inverter have to be considered to calculate more detailed results.

The DC voltage shown in Figure 4 and Figure 5 is calculated by the following expression.

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.

Figure 6 shows the frequency analysis results of a typical Fast Fourier Transform (FFT) of DC current ripple, id_ripple(t), for the 50 Hz/60 Hz case shown in Figure 4(a). In Figure 6, the contribution from the 50 Hz side has frequencies at 600 Hz (12th) and 1200 Hz (24th), and the contribution from the 60 Hz side has frequencies at 720 Hz (12th) and 1440 Hz (24th). Therefore the total DC current ripple, id_ripple(t), which appears on the DC side as shown in Figure 6(a), is overlapped by the characteristic harmonic due to the 50 Hz AC source with a characteristic harmonic due to the 60 Hz AC source. Figure 7 shows the magnified waveform portion B in Figure 4(a),

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 Figure 4, Figure 8 shows the waveforms of the DC side in case of asynchronous interconnection between the rectifier at 60 Hz and the inverter at 50 Hz. However, this case is with different smoothing reactors on the rectifier side and inverter side. Firstly, as Figure 8 shows, the reductions in DC harmonic currents are worthy of close attention.

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 Figure 8, an impressive fact is found that the characteristic harmonic magnitude (600 Hz, 12th) of the 50 Hz frequency side is always greater than that of the 60 Hz frequency side. That is, the characteristic harmonic of the lower AC frequency side is always greater whether the lower frequency side has a rectifier or not, and is not related to the values of the smoothing reactor.

Figure 9 shows the simulated comparisons between the asynchronous interconnection (50 Hz/60 Hz) and the

synchronous interconnection (60 Hz).