Harmonic Susceptibility Study of DC Collection Network Based on Frequency Scan and Discrete Time-Domain Modelling Approach

Imbaquingo, Carlos Enrique; Sarrà, Eduard; Isernia, Nicola; Tonellotto, Alberto; Chen, Yu-Hsing; Dincan, Catalin Gabriel; Kjær, Philip; Bak, Claus Leth; Wang, Xiongfei

doi:https://doi.org/10.1155/2018/1328736

Journal of Electrical and Computer Engineering

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 1328736 | https://doi.org/10.1155/2018/1328736

Harmonic Susceptibility Study of DC Collection Network Based on Frequency Scan and Discrete Time-Domain Modelling Approach

Carlos Enrique Imbaquingo,¹Eduard Sarrà,¹Nicola Isernia,¹Alberto Tonellotto,¹Yu-Hsing Chen,¹Catalin Gabriel Dincan,¹Philip Kjær,¹Claus Leth Bak,¹and Xiongfei Wang¹

Guest Editor: Luigi P. Di Noia

Received19 Jul 2018

Revised31 Oct 2018

Accepted06 Nov 2018

Published16 Dec 2018

Abstract

The equivalent model of offshore DC power collection network for the harmonic susceptibility study is proposed based on the discrete time-domain modelling technique and frequency scan approach in the frequency domain. The proposed methodology for modelling a power converter and a DC collection system in the frequency domain can satisfy harmonic studies of any configuration of wind farm network and thereby find suitable design of power components and array network. The methodology is intended to allow studies on any configuration of the wind power collection, regardless of choice of converter topology, array cable configuration, and control design. To facilitate harmonic susceptibility study, modelling DC collection network includes creating the harmonic model of the DC turbine converter and modelling the array network. The current harmonics within the DC collection network are obtained in the frequency domain to identify the resonance frequency of the array network and potential voltage amplification issues, where the harmonic model of the turbine converter is verified by the comparison of the converter switching model in the PLECS™ circuit simulation tool and laboratory test bench, and show a good agreement.

1. Introduction

Offshore wind power is becoming an important energy resource in Europe. However, electrical power losses are always a concern in the operation of wind farm with long-range power transmission. To reduce losses, researchers look to replace AC with DC in the entire path from the wind turbine, through power collection and transmission to shore. Medium-voltage DC (MVDC) collection of wind power is an attractive solution to reduce overall losses and installation cost [1, 2]. The article [2] addresses cost-efficient solution to form the multiterminals of DC grid with the diode rectifier unit (DRU). However, the solution of DRU can produce a considerable harmonic current within the array network if the 6-pulse DRU topology is used.

Figure 1 illustrates a DC collection network of radial type of offshore wind farm with the HVDC transmission. This general configuration in Figure 1 supports any choice of MVDC collector voltage level as well as converter topology and the number of wind power plant (WPP) clusters. For example, the turbine and offshore substation can employ any DC/DC converter topology with multiple degrees of freedom: uni- vs bidirectional power flow; modular (parallel/series connection) vs monolithic; topology (single active bridge, dual active bridge, and more [3–5]). Figures 1 and 2 illustrate the details of wind farm configuration including wind turbine DC/DC converters, offshore substation converters, and onshore substation converters. The offshore wind farm is designed to generated and deliver 700 MW power from offshore to onshore [1].

The series resonant converter (SRC#) is selected as the turbine DC/DC converter based on the converter efficiency, high-voltage transfer ratio, scale of converter, and galvanic isolation property. Considering the existing converter topology and its power rating in high-power applications, the modular multilevel converter (MMC) with the galvanic isolation and unidirectional power flow control is selected to serve as the offshore substation converter to deliver the offshore wind power to the onshore grid via the HVDC transmission line. The onshore substation captures the power from the offshore wind farm by the onshore MMC with a grid-connected 50/60 Hz transformer. The onshore substation is designed to match with onshore AC grid voltage (utility) and deliver the offshore power to the onshore AC transmission network and alleviate the influence from the grid fault.

In this study, a series resonant converter topology (named SRC#) shown in Figure 3 is selected as the candidate for DC power conversion via ±50 kV_DC MVDC array network. With the series resonant converter, the DC turbine converter can take advantages of high efficiency, high-voltage transformation ratio, and galvanic fault isolation for different ratings of turbine generator [6–9]. A series resonant converter SRC# with the resonant tank on the secondary side and governed by the phase shift and a frequency-depended power flow control technique is here considered [10, 11].

To synthesize the DC-connected wind turbine converter and the MVDC gird, the wind turbine converter has to meet the following requirements:(1)Control of the LVDC bus(2)High-voltage transformation from the LVDC to MVDC(3)Galvanic isolation(4)Roust and compact design

Large amount of possible topology of converter topologies for a DC turbine is investigated. However, all of existing converter designs are immature technology, thus selecting an optimal topology is not a straightforward solution for converter design [12]. The turbine converter is designed to deliver the captured wind energy produced by the generator to the MVDC gird and then control the LVDC bus. Regarding power rating, number of components, and the possible solution of the turbine converter topologies in the early DC wind farm design, the hard-switched full-bridge power converter is selected as preferred topology [4, 13–17]. Therefore, the proposed series resonant converter # (SRC#) combined with the medium-frequency transformer (MF transformer) which serves as the DC turbine converter is selected to boot the low-voltage DC (LVDC) to high-voltage DC (MVDC-side of SRC#) with a high-voltage transferring ratio and provides a galvanic isolation to make sure that the line is immune from line faults [4, 18, 19].

The concept of SRC is widely used in the traction applications, which is usually operated at constant frequency and subresonant mode to archive a roust and compact design target in high-voltage specifications [20–24]. One of famous modulation techniques (or topologies) is the half-cycle discontinuous-conduction-mode series resonant converter (HC-DCM-SRC). Under the traction applications, the DC/DC converter is tied between two DC voltages with a fixed voltage transfer ratio and the open-loop control. Relying on the pulse removal technique, the bulky transformer in the classic SRC can be voided in the proposed SRC#, and thus the compact convert design can be archived [25]. More details of operation principle of SRC# is addressed Section 2.

To archive the study of harmonic susceptibility in offshore wind farm with the selected turbine converter, the analysis of array network in wind farm includes modelling the selected turbine converter (SRC#), array cable, and substation converter, where the substation converter in this study is considered as an ideal voltage source to produce DC component of MVDC and harmonic components. The objective of the study is to investigate how the harmonic affects the turbine converter and array networks. In the power collection system, the wind turbines and substation are connected together by medium-voltage DC cables. However, there are no standards or guidelines for such DC array network design, and thus some studies are still made by simply selecting commercially available medium-voltage AC submarine cables without giving any additional consideration in the cable’s properties such as insulation [26].

Modelling of the array cables is an essential task for evaluating results of cable sizing by computer simulation. The complexity of modelling is dependent on the phenomenon, which will be conducted in the study case. For example, the travelling waves (voltage and current) with a finite propagation speed cause reflections. This phenomenon has to be represented on the cable model when study cases are intended to conduct the overvoltage or switching transient issues. Typically, three types of cable models, Pi Model, Bergeron model, and frequency-dependent models, which are available in the electromagnetic transient simulation program PSCAD™ (EMTDC) for studying AC and DC power collection systems are used [27–30]. To adequately represent the cable parameters for the harmonic susceptibility studies over different frequency ranges, the frequency-dependent array cable models are generated from geometric cable data based on the method [31] and recommendations proposed by the articles [32] and verified by the generated cable model in PSCAD™.

Although these DC alternatives promise lower electrical losses and lower bill-of-material, it is still an immature technology. As the many existing AC wind power plants, the harmonic pollution is an important issue since wind power collection systems serve by large amount of turbine converters [33–36]. Depending on the configuration of the wind power plant, every power collection system has its own inherent resonance behavior. The network, which is rich in the harmonic current, can increase the power losses and the stress of power components and cause unpredicted equipment trip. The resonance behavior inside the collection network is affected by properties of the array cable, filter designs, converter topologies, and PWM schemes. Therefore, the objective of this paper is to develop a methodology to identify potential harmonic resonant problems in the DC collection network. For example, the phenomenon of harmonic amplification relates to grid contingencies, variation of load flow, cable length, and uncertainties of network [33, 37]. The article [38] focuses on the stability issues of DC grid related to the travel waves on the transmission line and harmonic content of converters. The paper identifies the travelling wave in the long submarine array cable and resonant problem between the cable length and the switching action of the substation converter. According to the practical cases, the DC wind farm topologies for the case study mainly focus on the offshore wind farm with a short DC power transmission path.

Reviewing the common techniques for wind farm analysis, the frequency scan techniques are usually used for identifying harmonic voltage distortion of specific harmonic component in wind farms [34, 39, 40]. In this case, the wind turbines are represented for a specific frequency with an admittance equivalent circuit. The array cable model is selected according to the cable length and interesting frequency ranges of harmonic analysis. To adequately represent the cable parameters in the study, both equivalent PI section and frequency-dependent phase model are generally used to consider the frequency dependence of the cable parameters [28, 31, 41]. Additionally, the wind farm configuration could be changed via the operation of disconnectors or circuit breakers due to maintenance or forced outages of fault. Therefore, the harmonic study of wind farm must include all agreed switching configuration [39].

2. Review of Operation Principle of Series Resonant Converter #

The circuit topology of series resonant converter # (SRC#) with the medium-frequency transformer is depicted in Figure 3, and the topology comprises a full-bridge inverter, a medium-frequency transformer, resonant LC tank, and medium-voltage output diode rectifier. The switch pairs T₁ (leading leg) and T₂ and T₃ (lagging leg) and T₄ operate at a 50% duty cycle as in Figure 4(e). Commutation of switches on the leading leg (T₁, and T₂) is shifted by an angle of δ, which is respect to the conduction of the switches on the lagging leg (T₃, and T₄). The duration of phase shift equals to half cycle of the tank resonant period, resulting in a quasisquare excitation voltage as seen in Figure 4(a), and then, the voltage passes through the medium-frequency transformer and excites resonant tank where the resonant tank current is given in Figure 4(b). Via the diode rectifier and the output filter, the current (or power) is delivered to the medium-voltage network, V_MVDC. Up to this point, there is no difference in the operation compared to a constant frequency and phase-shift control of classical SRC, which is usually operated in the super resonant mode, to achieve ZVS at turn on.

Considering the switching losses with IGTB applications, the subresonant mode is preferred in SRC# which allows ZCS or a low current at turnoff (Figure 5(a)); regardless of switching frequency, a full-resonant current pulse is delivered to the load [12]. The control law of SRC# is also allowed ZVS at turnon and ZCS at turnoff for the diode rectifier, as shown in Figure 5(b). It should be noted that the output power has a positive relation with the switching frequency, which means that the output power depends how much energy (current pulse) transfers to the output stage. SRC# operated in a low frequency means lower output power, while high-frequency operation will deliver a higher output power.

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3. Linearized Model and Closed-Loop Control of Series Resonant Converter

The objective of the study is to understand the harmonics distribution of offshore DC wind farm and how the DC wind turbines are affected by harmonics from the MVDC gird. Figure 6 summarizes the derivation of the plant model of the DC wind turbine based on the discrete time-domain modelling approach (Steps 1–8) [42], the transfer function of the output LC filter (Step 9), and SRC#’s control design (Steps 10–12). The details of the derivation (or equations) are address in Appendices A, B, and C which can help reader to reach the harmonic model of the DC wind turbine and then conduct harmonic susceptibility study of the DC wind farm. The following discussion including the flow chart only shows most important conclusions (or equations) of the derivation due to the limitation of page length.

The ideal of the harmonic model of the DC wind turbine converter is developed based on the harmonic model of the AC wind turbine and linearized model of the series resonant converter (SRC#) with discrete time-domain modelling approach [42, 43]. A linearized state-space model of SRC# in subresonant mode is reported in (1) and in (A.37) in Appendix:where and .

It is enough to recall here that the duty cycle is fixed to 50%, and one event is defined as a whole time interval in which the IGBT T₁ is pulsed or not, lasting half of the switching period [10]. The state variables are meaningful of the tank current and resonant capacitor voltage at the beginning of one event, while the rectified output current is averaged over one event. The average output current is regulated by the proper action of the switching frequency. Transfer functions between converter output current and input variables are

The transfer function between output current and input variables (i.e., ) can be obtained viawhere

Those transfer functions describe how the averaged output current of the converter, before the output LC filter, is influenced by input disturbances. Regarding the voltage harmonics in the array network (MVDC network), the transfer function can be used to evaluate the effect of the voltage harmonics on the converter output current. Detailed derivation of the linearized state-space model for SRC# from (1) to (3) and control design including the expression of elements in [A], [B], [C], and [D] matrixes are revealed in Appendix.

The output LC filter is designed with the natural frequency of 100 Hz (i.e., ≈ 100 Hz) in order to have constant power delivered to the MVDC grid and attenuate high frequency harmonics, around twice the switching frequency. Based on the plant model of SRC#, the transfer function of the compensator (or controller) in the control block shown in Figure 7 for a specific operating point (OP) is set as

The transfer function in (5) is designed based on predetermined specifications, such as the ability to reach its reference output current with least ringing, overshoot, and fastest dynamic during perturbations in input variables. The compensator is composed by an integrator to minimize the steady-state error, a pole around the resonant frequency of the filter to lower the resonant magnitude peak influence (), and finally a gain K, necessary to set a proper crossover frequency, which in this case needs to be lower than the resonant frequency of the filter to avoid instability. Control deign of SRC# is revealed in Appendix, and the parameters of the controller are described in Section 6 that operates in the working conditions specified in Table 1 are shown in Table 2.

Figure 8 demonstrates the transfer function at the operation point (ω_p = −250 rad/s, K = 50.8 dB, and P_out = 12.7 MW). The result shows that the gain margin G_M is infinite (>0 dB) and the phase margin Φ_M is around 78.7 deg. at 49.8 rad/s which represents that the controller is stable. The design criterion has been implemented to select the parameters for the different operation points (different output powers of the DC wind turbine), which have enough gain margin and the phase margin to deal with the parameter uncertainty and variation.

4. Modelling of DC Wind Turbine Converter for Harmonic Susceptibility Study

To identify the harmonic model of the DC wind turbine converter for the harmonic susceptibility study, the voltage harmonics within the MVDC network are considered as a perturbation on the steady-state operation point. According to the control block diagram of the SRC# given in Figure 7, the harmonic components (disturbance component ) on the output current are dominated by the disturbance on the power reference signal , the disturbance in the low-voltage DC network , and the disturbance in the output terminal voltage of diode rectifier .

Without the disturbance component in the power reference signal and in the low voltage DC network ( and ), therefore, the harmonic model of WTG (wind turbine generator) considering the output LC filter ( and ) can be represented by an admittance equivalent in (6) as given in Figure 9, where the harmonic admittance is affected by the circuit parameters of DC wind turbine converter topology (plant model), the control design, and the output LC filter:where the admittance of the converter without considering the output filter can be represented by the following equation:

Equation (6) describes satisfactorily the variation of the converter current delivered to the grid in response to grid voltage disturbances where the transfer functions and in (6) and (7) (the details of expression is in (3)) describe how the output current is influenced by the disturbances in the control input signal and the output voltage (∝ voltage V_MVDC), respectively. In practical applications, the output LC filter contains a tiny amount of the parasitic resistances which can alleviate the peak value of admittance magnitude at the resonances frequency, as done in the following sections for the study case described in Section 6.

Figure 10(a) shows the variation of admittance magnitude of the converter with/without the output LC filter. It also represents the magnitude of the converter without the output LC filter to observe the filter’s influence on the equivalent admittance of the converter. Admittance magnitude of the converter with the output filter is high as it approaches the natural frequency of the output LC filter (f_f ≈ 100 Hz), while the magnitude of admittance of the converter itself (without output filter) is low in the whole spectrum and reaches a constant value at low frequencies. Regarding the phase angle of the admittance equivalent, the behavior of the converter with output filter changes from inductive, at low frequencies, to capacitive, at the natural frequency of the filter as shown in Figure 10(b).

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5. Harmonic Susceptibility Study of DC Collection Network

In this section, the harmonic susceptibility study of DC collection network is revealed based on the harmonic model of the turbine converter developed previously and compared to a switching model implemented in PLECS™. According to Figure 9, a radial-type cluster of seven wind turbines of an offshore wind power plant (OWPP) is given. It will represent the study case for the comparison of the harmonic model and switching model. The parameters for the turbine converter and operating point of the plant model with controller are given in Section 6. Since different configurations of collection network and wind turbine converter topologies could generate different harmonic spectra, background harmonics of the DC grid (correction/array network) from the offshore substation DC/DC converter and DC wind turbine generator (DC WTG) can cause undesirable resonant phenomena and raise stability issues. One of the major tasks of the harmonic susceptibility study is to identify resonance frequencies of network and investigate potential voltage amplification issues.

5.1. Cable Model

Array cables in the following study cases are modelled as a pi-equivalent model with longitudinal admittance parallel branches in order to consider the frequency dependency of distributed cable parameters and represent correctly the behavior in the frequency ranging from 0 to 10 kHz [31, 44]. In the study, the electromagnetic transient program PSCAD is usefully employed to extract the longitudinal admittance and the shunt capacitance from the geometric data of array cable [41]. The values of resistance, inductance, and capacitance to use at each frequency are calculated by the software individually.

According to the cable model in PSCAD, the capacitance is not varying within the range of interesting frequency. Instead, as the frequency rises, the resistance keeps increasing, due to the skin effect, and the inductance decreases its value due to the eddy currents in the outer conductors. For the considered study case, eight RL series branches are used to approximate the longitudinal impedance.

The PSCAD generates the admittance date (i.e., matrix [Z_PSCAD] and [Y_PSCAD]) from the geometrical data of the marine cable (in frequency-dependent (phase) model) in certain operational frequency to form “pi-equivalent cable model” [45]. An equivalent pi-equivalent model is built based on the cable model, which can fit the admittance date (matrix [Z_PSCAD] and [Y_PSCAD]) from the geometrical data of the submarine cable (implemented in frequency-dependent (phase) model in PSCAD) to make sure that the adequate cable model is implemented in the array cable network. Actually, the frequency-dependent phase model in PSCAD is also regarded as a most numerically accurate cable model thus far. The formulation of models has been proved that the bandwidths of the model can be up to 1 MHz, which can satisfy most of transient studies and harmonics studies [41, 45].

5.2. Frequency Scan Approach

The WPP cluster consists of seven turbines with seven MVDC cable segments, where the offshore substation converter is represented by an ideal harmonic voltage source. According to the frequency scan approach for network analysis addressed in Figure 11, the harmonic voltage source (disturbance component) in the output DC voltage of the offshore substation converter is activated and injects harmonic voltage component into the MVDC grid with 0.5% (=0.005 p.u) of voltage magnitude ( = 500 V) and the frequency of injected harmonic voltage raises until 300 Hz, by steps of 20 Hz. The output voltage and output current traces of each wind turbine model in each frequency step are extracted for FFT analysis after the steady-state condition is reached. This approach allows an arbitrary choice of frequency and magnitude of the disturbance in the offshore substation. Therefore, it does not intend to represent only a specific range of harmonics in the DC collection network. Eventually, models of the DC collection system (or network) could be established for different frequencies of the injected disturbances. As an example, for the harmonic model, the wind turbine converter can be described by a particular RL or RC series branch for disturbance frequency.

Based on the harmonic model of DC collection network in Figure 9, Figure 11 gives a flow chart of frequency scan approach for the DC collection system in the harmonic susceptibility study. The process includes the identification of the harmonic model of the DC wind turbine for certain frequency of the injected disturbance (harmonic) and the establishment of the harmonic model of DC collection network. The DC collection network in Figure 9 is set as an example to explain the sequence of frequency scan illustrated in Figure 11. Step 1: the harmonic voltage source in the offshore substation () is connected to the network and injects a 20 Hz harmonic disturbance into the MVDC network with a specific voltage magnitude . The harmonic model of the DC wind turbine generator (WTG) under 20 Hz perturbation (harmonic frequency) in MVDC network is identified by connecting to the MVDC network at t = t₁. Step 2: the voltage and current within network are extracted for FFT analysis after the voltage and current have reached the steady-state condition. Step 3: at t = t₂, the frequency for which the harmonic model of the DC wind turbine is evaluated and is increased by 20 Hz, as the frequency of the injected disturbance. The algorithm turns back to Step 2. Eventually, the frequency sweep continues until 300 Hz (at t = t_n−1), which depends on the control bandwidth of the DC wind turbine converter and interesting frequency range in the harmonic susceptibility study.

6. Verification

Based on the DC collection network in Figure 9 and the converter control design addressed in (5), Tables 1 and 2 give the parameters used in the study of the DC collection grid, where the geometric parameters of cables are shown in Figure 12 which is implemented in the created equivalent PI section according to the frequency-dependent (phase) model in PSCAD. For the turbine harmonic model, all the turbines are considered operating in the same operating point, described in Table 2. As far for the switching model, wind turbine converters are operated with a switching frequency of 800 Hz that guarantees around an output power of 8.2 MW when the output voltage of the converter is 100 kV. The LVDC source (input voltage) of each converter is 4.04 kV, while the voltage drops across cables are responsible for different voltages at the output terminal of the wind turbines. The controller of each turbine regulates the switching frequency in order to control the output power to follow its reference. In particular, the frequencies are higher as wind turbines move further away from the offshore substation, to compensate the lower voltage difference on the two sides of the converter. According to the frequency scan approach illustrated in Figure 11, additionally, a perturbation with a specific sequence of frequency in the range of 20 to 300 Hz is applied to the corresponding harmonic model of turbine converter and the results are given in Figure 13 for a single wind turbine.

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Figure 13 gives a verification of the turbine harmonic model (equivalent admittance) by comparing the harmonic spectrum of output current with the results generated by the corresponding switching model of the turbine converter in PLECS™. The current spectrum shows that the derived harmonic model and converter switching model are very close up to 300 Hz. The magnitude of output current spectrum is reduced as the frequency of the harmonic component is far from the natural frequency of the output LC filter, where the mismatch between the harmonic model and its switching model in PLECS™ becomes higher. Furthermore, the turbine output current trace shows that the harmonic component of output current is amplified, while the frequency of MVDC harmonic voltage (V_MVDC,h,n) approaches to the natural frequency of output LC filter of the turbine converter.

The results of network (array network) simulation in Figure 14 between both models (harmonic model and the corresponding switching model) show a good agreement. However, some mismatch coming from the harmonic model developed can be seen in all turbines in cluster. Furthermore, as observed along the cluster, last wind turbines (shown in Figures 14(f) and 14(g)) have second- and third-order current harmonics due to nonlinear behavior considered by the turbine converter switching model. This is because the proposed harmonic model cannot predict these higher order harmonics since the modelling approach for the turbine converter in MVDC network are linear. Figure 15 gives a comparison of the harmonic spectrum of output current for wind turbines at different locations to study the error of the harmonic model. While in the closest WTG to the offshore substation, a sinusoidal wave is observed (WTG11); in the last ones, sinusoidal waves have a considerable amount of harmonic components at multiples of the injected disturbance frequency (WTG17). Furthermore, the amplitudes of the fundamental current harmonics decrease along the cluster, approaching the farthest wind turbine from the offshore substation. This phenomenon is related to the rising amplitude of higher order harmonics as depicted in Figure 15. Additionally, it is worth mentioning that, in the switching model, there are current harmonics around 1600 Hz, twice of the switching frequency of converter, as the filter at the output of each converter cannot cancel completely these current harmonics.

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The harmonic current spectrum at natural frequency of the output LC filter (≈100 Hz) was not shown in the grid simulation in Figure 14 because it was in the equivalent turbine model (admittance equivalent), since it was “ideal” current source (linear model), the parameters were really high and were not matching the switching model at al. Probably the filter damping is too small which results in very high current (approx. 500 A) at resonance of the output LC filter for a single turbine in grid simulation (equivalent model). However, the results of grid simulation still shows that the admittance equivalent follows results from the switching model for a single turbine in all frequencies; at natural frequency of the turbine output LC filter, the admittance equivalent is clearly not well represented.

In the verification of modelling of the array cable with equivalent PI-cable section based on the method proposed by Beerten et al. [31], the values of resistance and inductance of each branch are calculated in order to fit the behavior of the longitudinal admittance in the frequency domain calculated by PSCAD’s cable model (frequency-dependent (phase) model). Figure 16 shows a good matching of the fitted values of the admittance by the eight parallel branches () and the values calculated by software (PSCAD’s cable). The error on the magnitude of the admittance between two models is less than 1.0%, within the range of 0.001 Hz–10000 Hz. The error in phase is always less than 1.7°, in the range of interesting frequency.

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Since the calculation of the admittance date from the frequency-dependent (phase) model in PSCAD is not straightforward and is extremely complicate, there are some articles which are useful to conduct the calculation of admittance date from the geometry date of the cable model [28, 30, 45].

7. Laboratory Test

A laboratory test is performed to prove the validity of the proposed harmonic model of the turbine converter. A schematic of the setup of the DC wind turbine converter for SRC# is depicted in Figure 17. The down-scale prototype circuit with 216 V_DC (V_LVDC) to 400 V_DC (V_MVDC) DC wind turbine converter is built to verify the harmonic distribution of the turbine converter. The wind turbine converters, whatever AC or DC turbines, are the boost structures in order to increase the efficiency of power transmission. Therefore, the down-scale prototype circuit is designed to match with this character (boost the low-voltage DC, V_LVDC, from the generator) to match with the DC grid voltage (medium-voltage DC, V_MVDC) from the offshore substation. Although only a single turbine converter is implemented in the test bench, the harmonic signatures of the wind turbine converter (in the test bench) in the interesting frequency range have been confirmed which match with the simulation model as in the following results.

A programmable AC source is used to emulate the harmonic voltage source (V_MVDC,h) with sinusoidal waves at different frequencies on the output terminal of the converter. A module of resistors R_load and a diode D_Aux are placed on the output side of the converter to simulate the character of unidirectional power flow of MVDC network. Table 3 describes the power rating and circuit parameters of the laboratory test bench of SRC# where the average of output voltage is around 400 V_DC (V_MVDC = V_turb, DC component of output voltage of the turbine) and the output current (DC component of output current of the turbine) is around 1.4 A_DC.

A harmonic model and switching model in PLECS™ are also developed with the parameters of the laboratory setup for the comparison. The operating switching of SRC# is set to 800 Hz with 50% duty cycle and an output power of 550 W. Input voltage (V_LVDC) is set to 216 V, which is 11 V slightly higher than the input voltage (205 V) in the switching model developed in PLECS™ to take into account voltage drops in the practical setup. The DC component of converter’s output voltage (V_MVDC,0) is set to 400 V. A harmonic voltage source (V_MVDC,h) of 3.5V_RMS, with frequency varying in the range of 20 to 300 Hz, is used for harmonic susceptibility tests. Harmonics current in the output are recorded and the results from the laboratory experiment, PLECS™ simulations, and the admittance equivalent developed for the laboratory setup are compared in Figure 18. The results show that there is a good agreement with the proposed DC turbine harmonic model. However, there is a mismatch in the amplitudes (Figure 18(a)) when the injected frequencies approach the natural frequency of the output LC filter. As given in Section 4, the admittance of the DC turbine harmonic model is heavily influenced by the output LC filter. The behavior of the converter changes from inductive to capacitive at the natural frequency of the filter as in Figure 18(b).

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It should be noted that there is no standard for selecting the medium-voltage DC (MVDC) grid. The MVDC of 400 V_DC is chosen as output voltage of the down-scale prototype circuit of SRC# from the available test equipment in the laboratory for simulating the MVDC gird (controllable DC power source). In practical applications, the selection of MVDC is based on the electromagnetic transient (EMT) simulation of the grid and existing offshore wind farm technologies such as power converters, protection devices, available offshore platform, and array cables [1].

8. Conclusion

A model-based methodology in conducting the harmonic susceptibility study is revealed based on the discrete time-domain modelling technique and the frequency scan approach in the frequency domain. The developed harmonic model in the paper can be used to perform a harmonic susceptibility study of power collection network in an early stage of a wind farm project because of its simplicity and low computational cost compared to a switching model. In the later stage, the proposed method is recommended to consider the switching model of the converters in order to be closer to reality. The process of harmonic susceptibility study contains identification of the harmonic model of the turbine converter and the establishment of array network in the frequency domain. The validation of the harmonic model of the turbine converter is given in Sections 6 and 7 via computer simulation implemented in PLECS™ and the laboratory tests, respectively. Additionally, the distribution of current harmonics within DC collection network is investigated including the identification of possible resonance frequency within array network. Moreover, analytical data from simulation and the laboratory tests give indications to set a proper design and natural frequency for the output LC filter of the DC turbine converter and a matched switching frequency for the converter placed in the offshore substation.

Appendix

According to the flow chart of mathematical derivation in Figure 6 in Section 3, firstly, the circuit topology of the turbine converter and mode of operations are decided as shown in Figures 3 and 19, and then the equivalent circuit based on the switching sequence of transistors is created as shown in Figure 20. In Figure 20, the large-signal model of the converter is generated (Step 4) and then the interesting state variables (Step 5) are defined to create the small-signal equations. Steps 6 and 7 establish the space model of the plant. Finally, the converter plant mode is established based on the discrete time-domain modelling approach proposed in (Step 8) [42, 46]. Step 9 gives a set of interesting transfer function of the output LC filter. Eventually, Steps 10 to 12 focus on the control design for SRC#.

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A. Derivation of the Plant Model

Steps 1–3. Decide the circuit topology of DC turbine converter, resonant tank waveform, and equivalent circuit.

Steps 1–3 describe how to obtain the large-signal model and the corresponding equivalent circuit in subresonant CCM in Figure 20 from the circuit topology of SRC#, which is operated in subresonant CCM as in Figures 3 and 20, respectively. The waveform in Figure 19 is divided by different time zones (different switching sequences) based on the discrete time-domain modelling approach. The figures are eventually used to generate the large-signal model of SRC#.

Step 4. Large-signal model.

In the large-signal model, the final value of interesting state variables in each switching interval is represented by the initial values of states. According to discrete time-domain modelling approach, the procedure is only valid when the variations in MVDC grid voltage (output terminal voltage) and LVDC voltage (input terminal voltage) in the event (switching) are relatively small than their initial and final values [42]. According to Figures 19 and 20, Equations (A.1), (A.3), (A.4), (A.6)–(A.10), (A.12), and (A.13) show the details of derivation of the large-signal model of resonant inductor current , capacitor voltage , and their end values at k_th event in terms of initial values of k_th event.

For t_0(k) ≤ t ≤ t_1(k) (T₁, T₄ ON),where

The resonant inductor current and resonant capacitor voltage can be obtained by (A.1):where

At time t = t_1(k), the tank current reaches the point of zero crossing, commutating the IGBT switches T₁ and T₄ turnoff and turning on body-diode D₁, and IGBT switch T₃. Therefore, the resonant inductor current can be expressed bywhere

For t_1(k) ≤ t ≤ t_2(k) (D₁, T₃ ON),wherewhere

Eventually, the end values of inductor current and capacitor voltage (at time t = t_2(k)) can be written aswhere

The same process of derivation as given in (A.1), (A.3), (A.4), (A.6)–(A.10), (A.12), and (A.13) can be applied for obtaining the large-signal expression of resonant inductor current and resonant capacitor voltage in (k + 1)_th event (t_0(k+1) ≤ t ≤ t_1(k+1) and t_1(k+1) ≤ t ≤ t_2(k+1)).

A.1. Steady-State Solution of Large-Signal Model

The following equation expresses the conditions for obtaining steady-state solution (at certain operating points of CCM) of the discrete state equation of the large-signal model:

By substituting (A.15) into (A.12) and (A.13), the steady-state solution of and can be written in term of :where the overbar used in equations is to indicate the steady-state value of the interesting state variables. For simplifying the derivation of the plant model, the output LC filter of SRC# (i.e., and ) is neglected due to a very slow dynamics in voltage and current comparing with the dynamics in the resonant inductor current and resonant capacitor voltage . Therefore, only the DC component of the output current diode rectifier is considered as an output variable which is expressed by . Eventually, the output current equation delivered by the SRC# during the K_th event is express aswhere

It should be noted that the initial value of inductor current is represented by I_r,0(k), V_Cr,0(k) is the initial value of capacitor voltage, the initial value of rectifier output voltage is written as V_o,0(k), and V_g,0(k) is then the initial value of input voltage of SRC#. Characteristic impedance Z_r is defined by the parameter of resonant tanks (), α_k is the transistor and diode conduction angle during the switching interval event k, and the switching frequency of the converter is represented by θ_s (=ω_s·t). Finally, the steady-state solution of the discrete-state equation for the output variable can be obtained by substituting the steady-state condition into (A.17) as

Step 5. Define state variable.

Control design technique based on the linear control theory cannot directly be applied for, mainly because of the high nonlinearity in the discrete large-signal state equations in (A.12), (A.13), and (A.17). Therefore, the linearization of the above large-signal equations is necessary. Equation (A.20) shows the definitions of interesting state variables in both k_th switching event (t_0(k) ≤ t ≤ t_2(k)) and (k + 1)_th switching event (t_2(k) ≤ t ≤ t_2(k+1)) which is used in the linearization process. Eventually, the following equations of approximation of derivative are applied for converting the discrete large-signal model into the continuous time:where

Instead of using the state variables in (A.12) and (A.13), the state variables in (A.12) and (A.13) are replaced by the defined state given in (A.20). With the approximation of derivation in (A.21), the nonlinear state-space model is given bywhere the output equation is

Step 6. Linearization and small-signal model.

Inject a small perturbation in all the interesting state variables in Step 5 during the steady state (near the certain operating point, OP). The nonlinear state equations can be rewritten with Taylor series expansion in terms of the operating point (OP) and the perturbations:(i)Taylor series expansion of resonant inductor current:wherethenwhere the subscript in equations indicate that the equation is evaluated at that steady-state point:(ii)Taylor series expansion of resonant capacitor voltage:wherethenwhere(iii)Taylor series expansion of the output current equation:wherethenwhere

The linearized equations can be obtained by neglecting the higher order terms of perturbation signals and retaining only the linear terms in the Taylor series expansion for (A.27), (A.31), and (A.35).

Step 7. State-space model.

A set of the linearized state-space model (in (A.37)) of SRC# in the subresonant mode can be generated from (A.27), (A.31), and (A.35). Additionally, the expression of transfer functions between the input state variables and the interesting state variables are given in (A.44) and (A.46):

For derivative of equations and ,where

With the steady-state operating conditions,

Therefore,

The same derivation process, which are used to the derivatives of and with respect to input states and can be used to calculate the derivative of in the output equation and the derivatives of and with respect to input states , , and . Additionally, the angle and its steady-state solution can be calculated by the derivation of the large-signal model in (A.7) as

The derivative of with respect to input states , , , and at the given operating points is expressed by the following equation:where ω_rs (=ω_r/ω_s) is defined by the ratio between the natural frequency of the resonant tank (ω_r) and the switching frequency of SRC# (ω_s).

Step 8. Transfer functions.

According to (A.37), the transfer functions between the converter output current (output rectifier current) and input state variables can be obtained:whereand transfer functions between defined internal state variables and input state arewhere

The details of derivation of the linearized plant model and the expression of elements in matrixes [A], [B], [C], and [D] are shown in (A.37), where the expressions of transfer functions, , in (A.44) can be obtained by (3).

B. Derivation of Output LC Filter

Step 9. Derivation of transfer function of output LC filter.

The output current of SRC# delivered from the output diode rectifier to the MVDC grid has a high-harmonic content at twice the switching frequency and its multiples, and at the moment, only the mean value of the output current in one event is considered as the real output current. An output LC filter has to be placed on the output terminal of the converter in order to decrease as much as possible the current harmonics and mitigate the inrush current during fault. The following equations give the transfer functions of the output LC filter used to derive the harmonic model of the DC turbine converter:where and are the parasitic resistances of filters, and filter capacitance and filter inductance are represented by and , respectively. (=) is the MVDC-side voltage, is the output diode rectifier voltage of SRC#, and (=) is the LVDC-side voltage of SRC#. The dynamic model of the output LC filter in the control block is given by Figure 7.

C. Control Design of SRC#

Steps 10 and 12. Control design of DC wind turbine converter.

Main specifications for the design of the controller are addressed in articles [47, 48]. Since this is the first paper to address the closed-loop control of the DC wind turbine and then conductor to the susceptibility study of offshore DC wind farm, the paper did not provide specific numbers (/range) for control design of the DC wind turbine in the offshore wind farm. The following descriptions mainly focus on the identification of the coefficient of transfer function of the SRC# turbine converter, and thus it can give the readers a full freedom of control design related to the DC turbine in the offshore wind farms to meet the performance requirements such as stability, zero steady-state error, settling time, overshoot and ringing, and disturbances rejection capability.

According to the structure of the control block in Figure 7, the control deign of SRC# is influenced by the plant model and output LC filter where the open-loop transfer function is defined aswhere the current delivered to the MVDC grid is related with the switching frequency of the converter by the transfer function is represented by

The design of control loop starts by evaluating the frequency response of . At the resonant frequency (≈100 Hz) of the output LC filter, a peak on the magnitude occurs. The parasitic resistances of the capacitor and inductor can provide a limited damper to attenuate the resonant peak.

Following the specifications mentioned at the beginning of this subsection, the design of the controller starts by adding an integrator to minimize steady-state error with a step input. Thus, the output current will reach the reference value with a minimized tracking error after the transient.

Second, a pole around the resonant frequency of the output LC filter is added to lower the resonant magnitude peak influence of the filter.

Third, a gain is necessary to set a proper crossover frequency f_c which is always lower than the resonant frequency of the output LC filter; otherwise, the resonant peak will be present in the feedback loop and the phase margin of the open-loop transfer function will be less than 0°, leading to the instability of the closed-loop system. Finally, the generic expression of the transfer function of the designed controller can be expressed aswhere is the gain (Hz/A) of the controller and is a pole (rad/s) close to the resonant frequency of the output LC filter. In order to improve the transient response of the system and to reject harmonic disturbances, generally, the angular crossover frequency ω_c (=2π · f_c) of should be as high as possible. Typically, the selected crossover frequencies f_c for all the operating points, are always less than 10% of the switching frequency f_s (f_c < 0.1 f_s) in order to minimize the influence from the switching harmonics [48]. The crossover frequency of the open-loop transfer function , which can be assumed equal to the bandwidth of the system, characterizes the rapidity of the feedback system. Therefore, the higher the crossover frequency, the faster the response of the system is.

Nomenclature

:	Inductor in the resonant tank
:	Capacitor in the resonant tank
:	Resonant inductor current
:	Resonant capacitor voltage
:	Input voltage of the resonant tank referred to the secondary side of the medium-frequency transformer
:	Output voltage of the resonant tank
:	Low-voltage DC
:	Medium-voltage DC
:	Output terminal voltage of the DC wind turbine converter
:	Output current of the DC wind turbine converter ()
:	Output current of the diode rectifier of the DC wind turbine converter
:	Output voltage of the diode rectifier of the DC wind turbine converter ()
:	Inductor in the output filter
:	Capacitor in the output filter
:	Switching frequency of the series resonant converter defined by = ω_s/2π
:	Natural resonant frequency of the tank defined by .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2018 Carlos Enrique Imbaquingo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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