#### 3.2. Offshore Converter Control

The offshore converter is a hybrid converter that consists of a 6P-DR and a 2L-VSC for offshore converter stations. As the diode rectifier is uncontrollable, the VSC takes charge of all control aspects. In this research, the VSC controlled the offshore AC voltage and performed harmonic cancellation. The active power

P and reactive power

Q flowing into the OF-VSC are expressed as [

11]:

where

$\left|{V}_{OF-VSC}\right|$ and

$\left|{V}_{PCC}\right|$ are the voltage magnitudes of the OF-VSC and PCC, respectively.

$n$ is the voltage ratio of transformer

${T}_{vsc}$,

${\omega}_{pcc}$ is the angular frequency of the PCC voltage,

${L}_{vsc}$ is the reactance of OF-VSC, and

$\gamma $ is the phase shift between OF-VSC voltage and PCC voltage.

The block diagram of the hybrid converter control system is shown in

Figure 3. As seen in

Figure 3, the offshore wind farm is regarded as a single current source connected directly to the 33 kV offshore feeder. This simplification omits the voltage transformation between the low voltage at the terminal of the wind turbine (usually 690 V) to the offshore feeder voltage, however since the object of study of this research was not the wind farm itself but the hybrid offshore HVDC system, this simplification reduced the complexity of the simulation model.

The OF-VSC establishes the AC voltage and frequency at the wind farm PCC. Due to the connection of the 6P-DR transformer, the fifth and seventh harmonics appear in the current at the PCC, so the OF-VSC operates as an active filter to suppress these harmonics. As mentioned in

Section 1, the controller is developed using a synchronous reference frame (SRF). The output from the SRF controller and AC voltage controller results in the voltage reference for the SPWM.

#### 3.3. Harmonic Mitigation Using SRF-Based Controller

A considerable number of publications made use of proportional-resonant controllers to provide active power filtering capabilities to voltage source converters. The proportional-resonant (PR)-based controllers [

17,

18,

19] can follow the sinusoidal harmonic reference at their respective resonant frequencies by introducing an infinite gain at the desired frequency without a steady-state error. PR-based controllers can provide fast responses but their tuning is complex. This is because the controller resonant frequency should be well-tuned to the reference frequency, as the infinite gain band is narrow, making this control method susceptible to grid frequency variations. Non-ideal PR-based controllers use a second-order integrator, which produces a wider resonant frequency band but increases tracking error [

20].

The synchronous reference frames (SRF)-based controller [

18] applies several synchronous

dq harmonics frames and low-pass filters to detect the harmonics currents. The control signal passes through the low-pass filter on to the PI controllers. The advantage of using an SRF-based controller is that each harmonic component is transformed mathematically into two

dq DC signals, where easy-to-tune PI controllers provide stable control with no tracking error. In addition, an active power filter using an SRF-based controller is not affected by changes in grid frequency. The main disadvantage of an SRF-based controller for harmonics suppression is a slow response due to delays introduced by the low-pass filter [

20]. In this research, the SRF solution is preferred, given its robustness and easiness to deploy.

Figure 4 shows the current harmonics mitigation strategy in the hybrid converter based on SRF. The current is measured at the PCC, 6P-DR and OF-VSC. The harmonic currents of the 6P-DR are compensated by the 2L-VSC using multiple SRF control topology [

21].

The conventional process of detecting the current harmonics generated by the diode rectifier with the SRF-based controller is shown in

Figure 5. As seen in

Figure 5, the current harmonics are fed to different harmonic

dq transformations, which are rotating at particular harmonic frequencies (i.e., 5th, 7th, 11th, and 13th harmonic frequencies). The output of the harmonic

dq transformations is a signal that comprises a DC value and an AC value. The DC value represents the

d and

q components of the harmonic current, whereas the AC signal contains the rest of the harmonic components (and the fundamental AC signal).

The low-pass filter (LPF) aims to remove the AC part of the harmonic currents after being transformed by the harmonic

dq frame [

22]. The filtered

dq component of the harmonic currents is used as a reference for the different PI controllers of the control system. The controller action of the different PIs produces harmonic

dq modulator signals that will drive the OF-VSC to generate harmonic currents of the same magnitude but with a 180-degree angle shift to those found in the current coming from the diodes. As such, the harmonic currents from the diode rectifier will cancel each other out with the harmonic currents of the OF-VSC at the point of common coupling. To produce these harmonic modulator signals in the controllers of the OF-VSC, a

dq to

abc conversion is carried out using multiples of the synchronous frequency as its input. To ensure every harmonic current does not exceed the allowed range, every PI controller has a saturation limit. Without a saturation limit, the magnitude of the reference voltage of the VSC would be greater than the allowed voltage of the DC capacitor, which may result in saturation in the modulation signal to produce unwanted harmonic signals and would impact on the quality of the PCC current. This control approach can select to compensate individual current harmonics, according to the grid requirement, selecting a single or a group for compensation [

15]. In this study, the fifth and seventh harmonic currents of the 6P-DR had a greater impact on total harmonic distortion (THD), given its transformer connection. Therefore, these components were chosen for compensation.

Equation (6) below presents the generalized harmonics

dq transformation

T_{n} based on the SRF technique to achieve

dq components of the

n harmonics, where Γ

_{n} = sign [sin(2πn/3)] represents the sequence of the

n harmonics and is reflected as an algebraic sign.

The application of

T_{n} to a given three-phase AC signal results in DC

dq0 signals that represent the

n harmonic, plus an AC component which includes all the non-

n harmonics in the signal. These non-

n harmonics are translated into frequencies that depend on the sequence of the

n harmonic and the sequence of the rest of harmonics. Equation (7) shows the

d and

q harmonic currents calculation:

where

${i}_{n\_d}$ and

${i}_{n\_q}$ are the

n harmonics

dq signals, and

${\beta}_{n}$ is the phase shift of the

n harmonic,

${i}_{n}$ is the

n harmonic magnitude,

${i}_{k}$ is the magnitude of the

$k$ harmonic current,

${\omega}_{s}$ is the synchronous frequency in radians,

$t$ is the time,

${\beta}_{k}$ is the phase shift of the

$k$ harmonic, and

${\mathsf{\Gamma}}_{k}=\mathrm{sign}\left[\mathrm{sin}\left(2\mathsf{\pi}k/3\right)\right]$ represents the sequence of the

k harmonic and is reflected as an algebraic sign.

The dynamics of the

dq fundamental and harmonics currents between the VSC and the PCC point are given by,

where

$r$ and

$L$ are the equivalent resistance and inductance between the VSC and the wind farm,

${i}_{n\_d}$ and

${i}_{n\_q}$ are the average

dq current components of

$n$ harmonics,

${v}_{n\_d\_vsc}$ and

${v}_{n\_q\_vsc}$ are the

dq voltage components of

$n$ harmonics of average voltages by the VSC, and where

${v}_{d}$ and

${v}_{q}$ are the

dq voltage components of the PCC point. The transfer functions between

dq fundamental and harmonic currents and the

dq fundamental and harmonics voltage can be represented by the same equation if the grid voltage (

${v}_{d}\mathit{and}{v}_{q}$) and the cross coupling term (

${\omega}_{s}L{i}_{n\_q}\mathit{and}{\omega}_{s}L{i}_{n\_d}$) from Equations (9) and (10) are considered as disturbances. The relationships between the current and voltage in the

dq frame are shown in Equation (11):

As seen in Equation (11), the open-loop system has a stable pole at

$-r/L$, which can be cancelled by the PI controller.

$K{p}_{inv\_h}$ and

$K{i}_{inv\_h}$ are the proportional and integral constants of the

$h$ harmonic PI current controller. The value of

$K{i}_{in{v}_{h}}/K{p}_{in{v}_{h}}=r/L$ and

$K{p}_{in{v}_{h}}/L=1/{\tau}_{inv\_h}$, where

τ_{inv h} is the time constant of the close-loop system. Equation (12) shows a close-loop transfer function for the

h harmonic current controller, where

τ_{inv h} can be selected based on the desired speed of response of the closed loop system: