Design and control of a wind energy conversion system based on a resonant dc/dc converter
2013; Institution of Engineering and Technology; Volume: 7; Issue: 3 Linguagem: Inglês
10.1049/iet-rpg.2012.0069
ISSN1752-1424
AutoresShixiong Fan, Weiwei Ma, Tee C. Lim, B.W. Williams,
Tópico(s)Wind Turbine Control Systems
ResumoIET Renewable Power GenerationVolume 7, Issue 3 p. 265-274 ArticleFree Access Design and control of a wind energy conversion system based on a resonant dc/dc converter Shixiong Fan, Corresponding Author Shixiong Fan [email protected] China Electric Power Research Institute, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorWeiwei Ma, Weiwei Ma State Grid Smart Grid Research Institute, Beijing, 102200 People's Republic of ChinaSearch for more papers by this authorTee Chong Lim, Tee Chong Lim Electronic and Electrical Engineering Department, University of Strathclyde, Glasgow, G1 1XW UKSearch for more papers by this authorBarry Wayne Williams, Barry Wayne Williams Electronic and Electrical Engineering Department, University of Strathclyde, Glasgow, G1 1XW UKSearch for more papers by this author Shixiong Fan, Corresponding Author Shixiong Fan [email protected] China Electric Power Research Institute, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorWeiwei Ma, Weiwei Ma State Grid Smart Grid Research Institute, Beijing, 102200 People's Republic of ChinaSearch for more papers by this authorTee Chong Lim, Tee Chong Lim Electronic and Electrical Engineering Department, University of Strathclyde, Glasgow, G1 1XW UKSearch for more papers by this authorBarry Wayne Williams, Barry Wayne Williams Electronic and Electrical Engineering Department, University of Strathclyde, Glasgow, G1 1XW UKSearch for more papers by this author First published: 01 May 2013 https://doi.org/10.1049/iet-rpg.2012.0069Citations: 16AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract Recent wind energy harvesting technology development has focused on using dc collection and a possible smaller passive grid because of high energy efficient components in the converter system. This study evaluates and compares the hard-switched full bridge converter and the series–parallel LCC resonant converter, both with transformer coupling, and suitable for wind energy conversion system (WECS). Experimental results substantiate their merits and drawbacks in variable wind speed conditions. The unique, previously unexploited feature of the LCC resonant converter is increased gain at low output power, low speed and low input voltage. A hardware wind turbine simulator emulating an actual wind turbine is used, with its design and control briefly highlighted. This study demonstrates the feasibility of utilising an LCC resonant converter for WECS through simulation and experimental results, both at the same power level. Nomenclature A ratio of parallel to series capacitance, Cp/Cs B1, B2 maximum power points at different wind speed cp (cpmax) power coefficient (maximum) C LCC resonant converter tank capacitance Co converter output filter capacitance Cp (Cs) parallel (series) capacitance Gr overall gain of the LCC resonant converter Gt LCC resonant converter tank gain Iin (Iinref) converter input (reference) current Iinr rated value of converter input current Ipeak converter peak current k1 dc link voltage value k, k2 constant kp, ki parameters of proportion integration (PI) controller Lo converter output filter inductance Ls series inductance N transformer turns ratio n transformer primary winding turns Pin (Pout) converter input (output) electrical power Pinmax maximum converter input electrical power at different wind speeds Pinr rated converter input electrical power Pm mechanical power of the turbine captured from the wind Pmmax maximum mechanical power at different wind speeds QL (QF) load quality factor (at full load) Rac equivalent ac resistance of the LCC resonant converter RL (RLmin) load resistance (minimum) r blade radius S swept area Vp transformer primary voltage Vdc dc link voltage Vin, (Vinr) converter (rated) input voltage Vinref reference value of converter input voltage Vsec transformer secondary voltage Vw wind speed δ converter duty cycle λ tip speed ratio λopt optimal tip speed ratio η1 power conversion efficiency of the turbine/generator η2 LCC resonant converter efficiency ρ air density φ magnetic flux ωm rotor speed ω0, f0 (ωs, fs) resonant (switching) frequency Subscripts pu per-unit 1, 2 maximum power points (B1 and B2) value 1 Introduction The dc/dc converter is widely used in renewable energy source systems (wind energy and photovoltaic), and with fuel cells or battery systems etc. It is the preferred first conversion stage for the integration of renewable energy sources, in order to achieve multi-terminal dc transmission for interfacing multiple wind turbines. However most topologies have disadvantages. The conventional single-switch boost converter is one type of dc/dc converter which cannot achieve high voltage gain or handle higher power because of the output diode high current at high duty cycles and its recovery effects [1]. In addition, there is no electrical isolation. The hard-switched full bridge (HSFB) converter with an intermediate high-frequency transformer can be used for high-voltage, high-power applications. Compared to a 50/60 Hz transformer, a high-frequency transformer has the advantages of smaller volume and lighter weight. However, a high switching frequency causes significant switching loss in semiconductor devices. Thus there is a trade-off between switching frequency and device losses. In [2, 3], simulation based loss evaluation of three dc/dc converter topologies (HSFB converter, single active bridge converter and series–parallel LCC resonant converter) was performed. However, stresses and component selection were not studied. Moreover, wind energy conversion system (WECS) performance with a dc/dc converter was not demonstrated experimentally. For the HSFB converter, there is voltage overshoot and oscillation across the diode rectifier on the high-voltage output side, because of transformer leakage inductance and the output filter inductor [4]. This increases the number of series diodes for higher voltage sharing and also introduces electromagnetic interference problems. Jovcic and Ooi [5] presents a new step-up dc–dc converter with variable frequency control, without transformer isolation, for the integration of renewable sources. However variable frequency control complicates resonant components and output filter design. Resonant converters have been demonstrated as a feasible option for high-voltage power converters [6]. They allow high-frequency operation which results in magnetic component size reduction without decreasing converter efficiency. Thus, the series–parallel LCC resonant converter is proposed for used in a WECS. This paper is organised as follows. Section 2 illustrates a multi-terminal dc wind farm configuration, and compares the HSFB converter and the LCC resonant converter in terms of switch stresses and efficiency. Section 3 presents the operating principle of the constant frequency LCC resonant converter, and its design procedure based on maximum power point tracking (MPPT). System simulation and experimental results are presented in Section 4. 2 Multi-terminal DC wind farm system Fig. 1 shows a schematic of a multi-terminal dc wind farm with LCC resonant converters. The system consists of wind turbines, permanent magnet synchronous generators (PMSGs), diode rectifiers, LCC resonant converters and a centralised current source inverter (CSI) for grid connection. The individual wind turbines via LCC resonant converters are configured as current sources feeding the grid connected CSI, which is beneficial in stabilising the WECS. This configuration uses one inverter to deliver power to the grid to save on system costs. The dc-link voltage is controlled by the CSI with a voltage loop control. Fig. 1Open in figure viewerPowerPoint Multi-terminal dc wind farm with LCC resonant converters 2.1 Wind turbine aerodynamic characteristics The output mechanical power of the turbine, captured from the wind, is (1)The tip speed ratio (TSR) λ is defined as (2)Then the wind turbine output power (1) can be rewritten as a function of λ (3) The maximum wind power can be obtained if cp is maintained at its maximum value cpmax corresponding to the optimal TSR λopt for different wind speeds. The maximum mechanical power at different wind speeds is (4)k = (1/2)ρScpmax(r/λopt)3 is a constant and the aerodynamic terms are defined in the nomenclature. Fig. 2a shows the curves for wind turbine output power and maximum power against per-unit (pu) turbine speed at different wind speeds. Fig. 2b shows the power coefficient against TSR curve, which is used in the simulations and experimentation. Fig. 2Open in figure viewerPowerPoint Wind turbine power characteristics and cp at different wind speeds 2.2 dc/dc converter Each dc/dc converter in the WECS, connects a wind turbine to the dc link. Each tracks the maximum power by controlling the converter duty cycle during variable wind speed scenarios. Two dc/dc converter types, based on H-bridge topologies, are compared in terms of semiconductor stress and loss, and transformer loss and size. They are the HSFB converter and the LCC resonant converter dc-to-dc topologies shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint HSFB and LCC resonant dc/dc converter topologies a HSFB dc/dc converter b LCC resonant dc/dc converter It is assumed that the variables at rated wind speed are 1 pu, the dc-link voltage is k1 pu, and the cut-in wind speed is 1/3 pu. Thus the converter input voltage Vin ranges from 1/3 to 1 pu, when neglecting stator impedance and rectifier diode voltage drops. The rectifier bridge maximum output voltage is the H-bridge rated input voltage. For the HSFB converter, the transformer turns ratio N is designed for the minimum input voltage and maximum output voltage, to ensure the converter output voltage reaches the dc-link voltage. Thus N should be 1: 3k1, whereas for the LCC resonant converter, N is determined at rated input voltage. Assuming the LCC resonant converter tank gain Gt is 1, the transformer turns ratio is 1: k1. For both converters, the peak currents Ipeak occur at rated wind speed. The duty cycle is 1/3 for HSFB converter, and 1 for the LCC resonant converter. Owing to the 1 pu input voltage, the Ipeak of HSFB is 3 pu. For the LCC resonant converter, the H-bridge impresses a square wave voltage across the resonant tank, resulting in a sine wave current circulating within the H-bridge (with high-frequency current components attenuated by the high off-resonance tank impedance). Since the switching frequency is near the resonant frequency, the fundamental component (sinewave) of square voltage with amplitude of 4/π, is virtually in phase with the circulating current. Thus Ipeak of LCC resonant converter is 1.57 pu. For the HSFB converter, in order to reduce the diode rectifier side voltage spike, a conventional resistor–capacitor-diode (RCD) snubber circuit is used [7] or several energy recovery clamp circuits (ERCC) [8-10] have been proposed. Honnyong et al. [4] shows the diode rectifier peak voltage is 2.9 times the transformer secondary voltage Vsec without snubber circuits, but the diode rectifier peak voltage is still 1.3 times Vsec with the proposed ERCC. Since Vsec at rated input voltage is 3k1 pu, the diode rectifier reverse voltage rating is about 3.9k1 pu with the ERCC. For the LCC resonant converter, because of the sinusoidal voltage waveform at the transformer secondary side, the diode rectifier peak voltage occurs at the peak of the sinusoidal waveform, viz., 1.57k1 pu. There is no need for the snubber circuit to reduce the overshoot and loss problems. This also means, for a given transformer core area and flux level, more primary turns, 11%, are needed for the LCC case (LCC: Vp = 4.44nφfs, HSFB: Vp = 4nφfs), but because of the inherent voltage gain, the number of output winding turns is in fact reduced. The voltages across the HSFB converter and LCC resonant converter output inductor range from −k1 to k1 and −k1 to 0.57k1, respectively. The current through the inductor and output voltage are the same for both. The LCC resonant converter has short circuit protection capability because of the tank inductance. The series capacitor, acting as a dc blocking capacitor, in the resonant tank helps prevent transformer saturation. Table 1 shows comparative results based on a wind speed range of 1/3 to 1 pu. Table 1. HSFB and LCC resonant converters comparative results HSFB converter LCC resonant converter IGBT voltage 1 pu 1 pu IGBT peak current 3 pu 1.57 pu transformer turn ratio 1: 3k1 1: k1 transformer winding area 1 pu 0.9 pu (10% reduction) transformer saturation protection no yes diode bridge voltage high low diode bridge current same same short circuit protection capability No Yes Fig. 4a shows the duty cycle of the LCC resonant and HSFB converters based on the maximum power point (MPP) at different wind speeds. The HSFB converter duty cycle is inversely proportional to power at the MPP, from (33) in the Appendix, whereas the LCC resonant converter duty cycle δ depends on load quality factor QF and the ratio of switching frequency and resonant frequency fs/f0. When fs is close to f0, the duty cycle changes over a wide range, from power 1/8 to 1 pu. Fig. 4Open in figure viewerPowerPoint Comparison of duty cycle and efficiency between the HSFB and LCC resonant converters a Duty cycle of the LCC resonant converter (at full load QF = 2) and HSFB converter at the MPP, with different wind speed b Converter measured efficiency at the MPP with different wind speeds Fig. 4b shows the measured efficiency of the two converters at the MPP with different wind speeds. The efficiency measurement method for both converters uses two single-phase power analysers to measure the respective input and output powers (Voltech PM100, single-phase power analyser, 0.1% accuracy, dc-250 kHz bandwidth). The measured efficiency is Pout/Pin with input and output powers recorded simultaneously at thermal equilibrium, in a 22°C ambient. From Fig. 4b, the LCC resonant converter attains the highest efficiency at full load. Since the switching frequency is near resonance, the peak resonant current decrease is not drastic for larger QF, which means a decreased efficiency at light loads. For the HSFB converter, the H-bridge peak current decreases as the load current decreases, therefore it better light load efficiency. Owing to the lower duty cycle at full load power, larger peak current and lower efficiency occur. From Fig. 4b, the LCC resonant converter has better efficiency above 0.45 pu power, meaning the HSFB converter has better efficiency at light loads. Although currently, wind farms in operation have a capacity factor of < 50% on average, future development trends aim to greatly increase this, which is achievable by either improving the primary energy available – more accurate wind speed prediction and site location, or proposing optimal conversion technologies – better turbine characteristics and MPPT algorithms and control. The LCC resonant converter is a promising option in such future applications. 3 LCC resonant converter Resonant converters may be series-resonant, parallel-resonant and a combination of series–parallel resonant tanks. The series–parallel LCC resonant converter has the desirable characteristics of both series and parallel configurations [11]. For a WECS, the output power reduces as the wind speed decreases. In terms of dc-link output, the load decreases as the dc-link voltage is regulated constant by the CSI. The system needs higher gain because of an input voltage decrease as the wind speed decreases. The LCC resonant converter characteristic inherently fulfils this WECS requirement. It can provide high gain when the WECS operates at low output power, low speed and low input voltage. In contrast, the HSFB converter and pure series or parallel (and LLC) resonant converters, lower the dc gain when reducing the duty cycle at high power. This undesirable characteristic imposes a large electrical stress on the semiconductors. 3.1 Operating principles Fundamental mode approximation (FMA) analysis is used in the LCC dc characteristics design process. The input–output voltage transfer function of the LCC resonant converter is shown in (26) in the Appendix. Fig. 5a i shows LCC converter dc gain variation with different ratios of parallel Cp to series Cs capacitance, A, at load quality factor QL = 2. The dc gain increases as A increases. Fig. 5a ii shows dc gain against load quality factor QL with different A at fs/f0 = 1. The figure indicates that the LCC resonant converter can provide high gain as QL increases (load decreases – lower turbine speeds). Fig. 5Open in figure viewerPowerPoint DC gain characteristics and control of the LCC resonant converter a DC gain versus fs/f0 and QL with different A = Cp/Cs: (i) fs/f0 and (ii) QL b Control diagram of modified OPP control 3.2 Control and design methods Conventional phase-shifted pulse-width modulation at a constant frequency [12] is utilised to control the LCC resonant converter, which then allows a simple output filter and transformer design. The control and design methods of the LCC resonant converter are as follow. 3.2.1 MPPT From (4), if the power conversion efficiency of the turbine/generator is η1, then the maximum converter input electrical power Pinmax at each wind speed can be expressed as (5)where k2 = η1k. The electrical power is transferred to the grid by the intermediate dc/dc converter. The electrical power at the dc/dc converter input side is (6)To simplify the expression, (5) and (6) can be normalised, in the pu system as (7)For diode rectifier based WECS, Vin is proportional to ωm, in pu [13] (8)From (7) and (8), the average input current at the MPP is (9) The average input current is proportional to the square of the input voltage. Once the MPP status Vinref, Iinref of the WECS for a specific wind speed in the working range is obtained, the curve of input current at MPP against input voltage can be derived. The wind turbine utilises the curve to track the MPP. In this paper, modified one-power-point (OPP) control is used, as shown in Fig. 5b. The input voltage is measured, the corresponding input current is calculated according to (9), and then proportion integration (PI) control is employed to track the maximum power. To achieve MPPT, the LCC resonant converter needs to provide high dc gain when the wind speed changes, otherwise the wind turbine may only operate on the right-hand side of the power characteristics curve shown in Fig. 2a. In Fig. 5a, the LCC dc gain becomes insensitive when the switching frequency is far from the resonant frequency, whence the LCC resonant converter would not operate at the MPP. Thus the switching frequency should be close to the resonant frequency. The overall gain of the LCC resonant converter Gr is a combination of the resonant tank gain Gt and transformer turns ratio N(10)Fig. 2a shows two MPPs, B1 and B2, at different wind speeds. Gt,1, QL,1, Vin,1, Pm,1 and Gt,2, QL,2, Vin,2, Pm,2 represent the resonant tank gain, loaded quality factor, LCC resonant converter input voltage and mechanical power corresponding to B1 and B2. The LCC resonant converter voltage equations are (11) (12)therefore (13) According to (8) (14) If the LCC resonant converter efficiency is η2(15) (16) Substituting (15) and (16) into (14) gives (17) Substituting (30) in the Appendix into (17) gives (18)The resonant tank gain at the MPP, with different wind speeds, can be obtained from (19) Fig. 6 shows a 3D plot of the dc gain at different switching frequencies and QL with A = 1. Figs. 6a and b compare the resonant tank gain at the MPP with different wind speeds and LCC resonant converter dc gain at different QL values (full load), respectively. The lower surface (red) represents the resonant tank gain at the MPP, whereas the upper surface (blue) represents the LCC resonant converter dc gain that can be produced. When the resonant tank gain at the MPP is larger than the maximum dc gain at the corresponding load, the wind turbine cannot operate at the MPP. Thus the LCC resonant converter needs to develop gain to enable the wind turbine to operate at the specified MPP. Both figures indicated that a wide switching frequency range is allowable when a full load small QL is used. Fig. 6a shows the QL is between 2 and 20, whereas Fig. 6b shows the QL is between 8 and 80, both from 10 to 100% full-load. The switching frequency can be between 1 and 1.03 pu with small QL. Although for larger QL, the switching frequency which can achieve MPP wind turbine operation should be close to the resonant frequency. Fig. 6Open in figure viewerPowerPoint 3D plot of dc gain at different (a) Switching frequencies and (b) QL a Full load (at QL = 2) b Full load (at QL = 8) 3.2.2 LCC resonant converter design FMA analysis is used for the LCC resonant converter design, where in [14] it was verified that FMA analysis gives a reasonably accurate LCC converter design procedure. In [15], it was established that operation at the resonant frequency is preferable in minimising the resonant current, thus reducing electrical stresses. In this study, the LCC switching frequency is slightly above the resonant frequency, which maintains a high dc gain with zero voltage switching. The FMA analysis equations for the LCC resonant converter are derived in [14, 16]. Marian et al. [17] establishes the necessity for QL > 2.5 in order to provide near sinusoidal waveforms for accurate FMA analysis. It also shows the relationship between the minimum A and a given Gt, which gives the basis for choosing the resonant tank parameters. For convenience, the A is often chosen as 1 as a good compromise between bandwidth, component sizing, electrical stresses and impedance matching [18]. The design procedure is summarised as follows: 1. Calculate the Gr and minimum load resistance RLmin of the LCC resonant converter at rated power (20) (21) 2. Determine the resonant tank parameters: Selecting A = 1, where the LCC dc characteristics with A = 1 are shown in Fig. 5a, where the dc gain increases as QL increases. For a given QL, which corresponds to the minimum load resistance, using (26), Gt and fs/f0 can be obtained. The plots in Fig. 5a provide a design guide for fs and Gt. Then from (10) and (20), the transformer turns ratio N is obtained. Substituting (21) into (30) gives (22) After choosing fs, substituting f0 into (27) gives (23) Ls and C are obtained by solving (22)and (23) (24) (25) By substituting A into (28) in the Appendix, Cp and Cs are obtained. Gilbert et al. [16] gives a method which doubles the output voltage without minimum load change. Besides doubling the transformer turns ratio, the parallel capacitance quadruples and resonant inductance is quartered, doubling the output voltage while the switching frequency is unchanged. For the WECS, the LCC converter output minimum load is varied according to the dc voltage and power. Equations (20), (21), (24) and (25) indicate that the dc-link voltage can be changed to a different level, via the transformer turns ratio without changing the resonant tank parameters or switching frequency. 4 Simulation and experiment results 4.1 Test system description To verify the functionality and feasibility of the proposed system, a critical wind speed step change is applied to the proposed system, both in MATLAB simulations and experimentally. The turbine simulator, a vector controlled induction motor drive, driving a five-phase PMSG though a step-down gear-box, is controlled by an Infineon DSP Tricore 1796B, while the LCC resonant converter is controlled by micro-controller, DSPIC30F2020. The CSI (being decoupled by the dc-link inductance) can be modelled by a dc-chopper circuit in parallel with a shorting switch that maintains the average dc-link voltage constant by controlling the duty cycle of the two switches, as in Fig. 7a. The two switches are controlled by complementary gate signals with overlap. Fig. 7Open in figure viewerPowerPoint Test system a Schematic of the WECS b Experimental rig c Control flow for the turbine simulator The experimental WECS is shown in Fig. 7b. The turbine simulator characteristic parameters and LCC resonant converter parameters are shown in Tables 2 and 3. The control flow chart for the turbine simulator is shown in Fig. 7c. Table 2. Turbine simulator characteristic parameters Parameter Value air density 1.220 kg m−3 blade radius 2.86 m pitch angle 0° maximum power coefficient 0.4382 optimal tip speed ratio 6.335 gear-box ratio 4.88 driver type emerson unidrive PMSG stator resistance 2 Ω PMSG stator inductance 35.42 mH Table 3. LCC resonant converter simulation and experimental parameters Parameter Value parallel capacitance, Cp 73.3 nF (3×220 nF series) series capacitance, Cs 73.3 nF (3×220 nF series) series inductance, Ls 493 µH transformer turn ratio, N 1:1 resonant frequency, f0 37.4 kHz switching frequency, fs 38.8 kHz fs/f0 1.03 load quality factor at full load, QL 0.7 equivalent ac resistance at full load, Rac 83 Ω minimum load resistance, RLmin 67 Ω ratio of parallel to series capacitance, A 1 LCC resonant converter tank gain, Gt 1 output filter inductance, L0 4 mH output filter capacitance, C0 10 nF dc-link voltage, Vdc 200 V converter input voltage, Vin 66–200 V rated converter input electrical power, Pinr 600 W 4.2 Simulation results Fig. 8 shows the simulated mechanical and electrical performance of the WECS with a wind speed step change (a step change is used in order to determine the system dynamic characteristic limits). The wind speed is 3.5 m/s during the initial 1 s, then steps up to 4.2 m/s at t = 1 s. After 2 s, the wind speed steps down to and is maintained constant at 3.5 m/s. The power coefficient cp is maintained around the maximum value 0.4382. The rotor speed and converter input voltage and current are changed to track maximum power from wind. The dc-link voltage is maintained constant by controlling the duty cycle of the load side dc chopper. The output current depends on the power that is extracted by the wind turbine. Fig. 8Open in figure viewerPowerPoint Simulation results of the WECS for wind speed changes a Dynamic response of mechanical system b Dynamic response of electrical system 4.3 Experimental results Fig. 9a shows the dynamic response of mechanical system, including wind profiles, power coefficient, rotor speed and mechanical power. Fig. 9b i shows the input voltage and current waveforms of the converter, whereas Fig. 9b ii shows the dc-link voltage and output current. Fig. 9Open in figure viewerPowerPoint Experimental results of the whole WECS for wind speed changes a Mechanical system dynamic response b Electrical system dynamic response The simulation and experimental results both confirm the expected performance. 5 Conclusion In this paper, the HSFB and the LCC resonant converters have been compared in terms of switch stress, efficiency etc. The LCC resonant converter has better efficiency than the HSFB converter above half load conditions. The LCC resonant converter is proposed as an interface to a wind energy electrical source. Design procedures and analysis of the LCC resonant converter for a WECS has been presented based on FMA. The LCC resonant converter is designed at rated power, but provides increasing dc gain as the wind speed decreases without having to increases the transformer turns ratio, which is necessary for other converters topologies. WECS performance was investigated and analysed through simulation and experimentation, which demonstrated that the LCC resonant converter is a feasible option for WECS. 7 Appendix The LCC resonant converter input–output voltage transfer function is (26) The resonant frequency is (27)where (28) The ratio of parallel to series capacitances is (29) The loaded quality factor is (30) For a continuous conduction mode, the relationship between the HSFB converter input and output voltages is, pu (31) Equations (31) and (8) give (32) Substituting (32) into (7), and eliminating ωm,pu, gives (33) 6 References 1Mohan, N., Undeland, T.M., Robbins, W.P.: ' Power electronics converters, applications and design' (John Wiley & Sons, 1995) 2Max, L., Lundberg, S.: 'System efficiency of a DC/DC converter-based wind farm', Wind Energy, 2008, 11, (1), pp. 109– 120 (doi: https://doi.org/10.1002/we.259) 3Max, L.: ' Design and control of a DC collection grid for a wind farm'. 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