# An FTC Design via Multiple SOGIs with Suppression of Harmonic Disturbances for Five-Phase PMSG-Based Tidal Current Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Modeling and Problem Formulation

- Swell effects and turbulences are not considered for the tidal current speed;
- Only the 1st and 3rd harmonic components in back-EMFs are considered;
- Harmonic disturbances are discussed under a single-phase open case;
- The permanent magnets are surface mounted on the rotor of the five-phase PMSG used in the simulation setup and the laboratory prototype;
- Magnetic curve is linear;
- Eddy currents, iron losses are negligible;
- Stator phase windings are star-connected and neutral wire is absent.

#### 2.1. Model of Tidal Current Turbine

_{tides}means tidal current speed of, and r indicates the radius of turbine blade. Cp, the power coefficient of the turbine, depends on the tip speed ratio (TSR, λ=ω

_{tur}r/ν

_{tides}) and pitch angle β, where ω

_{tur}represents the angular velocity of tidal current turbine. Here, the Cp also suffers from an upper bound (around 59.3%) through such a horizontal-axis tidal current turbine by Betz’s law [27]. A three-dimensional Cp graphics is thus depicted as shown in Figure 2, which can be calculated as Equation (2).

_{1}~C

_{6}are coefficients of Cp (λ, β) function, which are given in Table 1. λ

_{ii}represents the intermediate variable of the TSR.

_{opt}(6.545) and Cp

_{max}(0.44335).

_{tur}, whose operational characteristics can be shown in Figure 3 (T

_{tur}-ω

_{tur}). It can be obvious to see that the optimal torque curve doesn’t pass through the points of maximum torque. The reason is that the maximum torque points correspond to the maximum powers, while the optimal torque is equal to that the maximum power divided by the mechanical speed as described in Equation (3).

_{tur}can be given by:

#### 2.2. Mechanical Model of Drive Train

_{gear}(N

_{gear}= ω

_{tur}

_{/}ω

_{m}), where ω

_{m}represents the mechanical angular velocity of the generator. It is worth knowing that the equivalent gear ratio in the small power-scale laboratory prototype is greater than 1 with the tidal current turbine. The real power-scale simulation setup makes use of a gearless type (N

_{gear}= 1). The mechanical coupling structures of the drive train are shown in Figure 4a,b. Particularly, the torsional damper is used to reduce variations of rotated shafts. The rigid coupling shafts and gears maintain the consistent angular velocities and angles of the inputs and outputs while there is a non-negligible angle shift (defined as θ

_{Δtd}) of the torsional dampers with flexible coupling structures.

_{tur}and J

_{gen}represent the inertia of the turbine and the generator, respectively. B

_{tur}, B

_{td}and B

_{gen}represent friction coefficients in the turbine’s shaft, torsion damping coefficient and the generator’s shaft, respectively. Mechanical and electromagnetic torques of the generator are denoted as T

_{m}and T

_{em}. The torque term T

_{tur}

_{1}is a discounted one relative to T

_{tur}, passing the turbine shaft, where T

_{tur}

_{1}= T

_{tur}

_{2}= N

_{gear}T

_{m}. In the second sub-equation, K

_{td}is the strength of anti-torsion for the torsional damper. It should be pointed out that the B

_{tur}, B

_{td}and B

_{gen}represent the lumped frictions by using slide bearings or roller bearings. For the gearless or so-called direct driven type in Figure 4b, the only difference is to remove the gearbox, whose model can be obtained as a special case in Equation (5), substituting N

_{gear}by 1. In this paper, the effects of torsional dampening are neglected, assuming the shafts are well coupled to each other.

#### 2.3. Modeling of Five-Phase PMSG in Healthy Conditions

_{x}, e

_{x}and i

_{x}(x = a, b, c, d, e) are, respectively, the phase voltages, back-EMFs and phase current. L

_{s}is the self-inductance of the stator windings. M

_{1}and M

_{2}are mutual inductances of the adjacent and non-adjacent phases, respectively. In this paper, the resistances of five phase stator windings are assumed the same for each the winding as R

_{s}. The back-EMFs of the generator contains 1st and 3rd harmonic components, which can be expressed as:

_{m}is mechanical angular velocity of five-phase PMSG. Φ

_{1}and Φ

_{3}represents the magnet fluxes yielded by the stator windings corresponding to the fundamental and 3rd harmonic components of the back-EMFs, respectively. φ

_{1}and φ

_{3}are defined as the corresponding initial phase angles. By a generalized Concordia transformation depicted in Figure 5, the five phase PMSG can be decomposed into three sub-generators, the principal, the secondary and the homopolar ones corresponding to three sub-spaces. Since the neutral wire is not connected and the stator windings are star-connected, the last sub-generator will be not available in healthy mode. In the principal sub-space, the original, Concordia’s and Park’s frames can be indicated by different axes with (a

_{pr}, b

_{pr}, c

_{pr}, d

_{pr}, e

_{pr}), (pα, pβ) and (pd, pq). Similarly, the three frames in the secondary sub-space are expressed by the axes (a

_{se}, b

_{se}, c

_{se}, d

_{se}, e

_{se}), (sα, sβ) and (sd, sq), respectively.

_{pr}and L

_{se}represent equivalent inductances for principal and secondary sub-generators, respectively. v

_{z}, e

_{z}and i

_{z}(z = pd, pq, sd, sq) are respectively phase voltages, back-EMFs and phase current under Park’s frames.

#### 2.4. Modelling of Five-Phase PMSG with Single-Phase Open

_{a}in the opened phase is expressed as:

_{a}. Then, the back-EMFs under the single-phase open condition can be expressed as:

#### 2.5. Additional Harmonic Disturbances Subject to Single-Phase Open

_{t}represents the conversion gain from torque to current references. The references in Equation (18) are convenient for controlling principal and secondary sub-generators.

_{b}

_{h}means the amplitude ratio of hth harmonic order in the current of phase “b”. θ

_{bh}is the phase shift of it relative to the phase angle in healthy conditions. The definition rules are the same in other phases. It should be noted that the phase shifts are constant and small in steady states, which are ignored here.

## 3. Multiple SOGIs-Based Model-Free FTC Strategy

#### 3.1. Single SOGI and Multiple SOGIs

_{sogi}and ω

_{sogi}represent the filter gain and its resonant frequency, respectively. i

_{α}(t) and i

_{β}(t) are output signals of the single SOGI.

_{sogi}and resonant frequencies ω

_{sogi}. It can be observed from Figure 7a,b that the bandwidths of both F

_{α}(s) and F

_{β}(s) are sensitive to the parameter K

_{sogi}. A greater value of K

_{sogi}brings a broader bandwidth around the resonant frequency but reduces the robustness of the SOGI [31]. On the contrary, a small K

_{sogi}will narrow the bandwidth and slow down its dynamic responses. In this paper, K

_{sogi}is fixed as $\sqrt{2}$ concerning the dynamic transient behaviors of the single SOGI. As the angular velocity of the input signal (ω) is equal to the resonant one of the single SOGI (ω

_{sogi}), i

_{α}(t) and i

_{β}(t) can be regarded as an estimated term and an orthogonal version of the input signal, respectively. According to Figure 7a,b, the single SOGI is able to estimate a certain frequency as ω

_{sogi}varies.

_{α}(t) is selected since it can estimate the input signal at a specific harmonic without phase shifts. However, for the concern of extension towards multiple harmonic orders, the above single frequency resonant based filter is clearly inadequate. As mentioned, a SOGI presents a stable characteristic for the selection of any frequency order from the input signal as shown in Figure 7a,b. A straight way for designing a multiple harmonic resonant structure is to establish a series of SOGI filters with different resonant frequencies in parallel. Similar to a single SOGI, the expression of multiple SOGIs [18] concerning on the lth order frequency can be organized as below using the transfer function F

_{α}(s).

_{α}

_{,j}(s) represents the transfer function of a single SOGI with the estimation of the jth harmonic resonance. n means the number of harmonic orders, which is a positive integral term.

#### 3.2. Multiple SOGIs-Based Model-Free FTC Design for Five-Phase PMSG-Based TCECS

_{pwm}/(1+1.5T

_{pwm}s). The time constant of switching frequency and the equivalent gain of pulse width modulation (PWM) are defined as T

_{pwm}and K

_{pwm}, respectively. V

_{carrier}, the amplitude of carrier signals, is set as 1. The controller parameters [15] in inner current loops can be given by:

_{Ω}and Ki

_{Ω}.The classical control law of the dual-loop system can be expressed as:

_{em}

_{1}and T

_{em}

_{3}represent the torque components with respect to the two power-generation sub-systems, which are respectively affected by the 1st and 3rd harmonics of the back-EMFs.

_{pq}and i

_{sq}in inner control loops by the description from Equation (27). Thus, the harmonic compensation by multiple SOGIs is then integrated only in the two q-axis control loops. In order to keep the level of T

_{em}, the estimate term of the DC component through the multiple SOGIs will not be injected to the i

_{sq}and i

_{pq}control loops. In detail, the extracted harmonics from output control commands v

_{pq}

^{*}and v

_{sq}

^{*}are injected with self-feedback loops. It should be noted that the parameter variations are neglected here, such as the inductances illustrated in Equation (14). d-axis current control loops are expected to be controlled within zeros, where the disturbances are not currently considered as the q-axis current control loops determine the main amplitude of torque under the vector control framework. The e

_{pq}and e

_{sq}contain even order harmonics, as per the analysis in Section 2.2, which are considered as external disturbances that need to be compensated together with the fault harmonic disturbances as the single-phase open fault occurs.

_{h_pq}and v

_{h_sq}represent the injected harmonic disturbances to the control commands v

_{pq}

^{*}and v

_{sq}

^{*}in feedforward paths, respectively. The corresponding compensation gains are denoted as K

_{h_pq}and K

_{h_sq}.

**Figure 9.**Control diagram in q-axis current control loops with the multiple SOGIs-based compensations.

## 4. Simulation Test by Real Power Scale Tidal Current Turbine Systems

_{tides}= 2.055 m/s). The other practical concern is that actual maintenance requests the implementation within the shortest possible duration.

#### 4.1. Comparison of Performance Using Five-Phase and Three-Phase PMSG

#### 4.2. Test of Fault-Tolerant Performance

- Under small third harmonic fluxFigure 12a,b present the simulation test results of the machine side, DC bus voltage and grid side when Φ
_{1}is 30 times more than Φ_{3}under operation at 0.393 MW. The fundamental frequency of the grid side is 50 Hz. With appropriate initializations, the whole system can reach the controlled mode at 0.04 s. At 0.06 s, phase “a” is set as an open circuit. At 0.1 s, the mSOGIs-based compensator for v_{pq}^{*}is activated. The other compensator for v_{sq}^{*}is then introduced after 0.05 s. From the first and second subplots in Figure 12a, it is clear that the speed fluctuations and torque ripples are suppressed via the compensators. The compensator for v_{sq}^{*}is used to constraint the harmonic disturbances in the secondary sub-generator, which is helpful in strengthening the fault tolerance combining with the compensator for the principal sub-generator. In the process of compensations, the perturbated degree of the control commands v_{pq}^{*}and v_{sq}^{*}are also reduced, according to the last subplot in Figure 12a, so as to directly reconfigure the PWM drive signals for the machine-side converter. However, the voltage command v_{sq}^{*}in the last subplot in the Figure 12a is much lower than v_{pq}^{*}in healthy conditions, while the fluctuation of v_{sq}^{*}is greater than v_{pq}^{*}once the single phase is open. Too great compensation for v_{sq}^{*}will worsen the suppression behaviors against the fluctuations, which will result in more perturbations in other control loops under the dual-loop control framework, such as the unconsidered d-axis control loops. That is to say, perturbations can “transfer” into other control loops. Consequently, the compensation gains are finally set as K_{h_pq}= 1 and K_{h_sq}= 0.1.In DC bus and grid side, as Figure 12b, the DC voltage, phase current as well as the output power of the grid-side converter (GSC) also perform attenuations of fluctuations under the single-phase open condition as the compensators are put into use. In detail, the power conversion processes create delays in the starting stage through the MSC and GSC. In steady states under healthy conditions, the powers in machine side, DC bus and grid side are almost equal. With compensations in faulty conditions, it is obvious that the fluctuations of power are also constrained. This is beneficial from the adopted fault-tolerant compensators in machine side. As a result, the proposed mSOGIs-based compensators are able to maintain the performance of the whole back-to-back conversion chain.**Figure 12.**Simulation result of compensation in q-axis current control loops when Φ_{1}/Φ_{3}= 30: (**a**) machine-side waveforms; (**b**) DC bus voltage and grid-side waveforms. - Under significant third harmonic fluxIn Figure 13, the magnet flux Φ
_{1}is nine times of Φ_{3}, which means that the third harmonic component in the back-EMFs is more significant in both healthy and faulty conditions. Fluctuations in machine side and grid side are well compensated, and their performance analysis is consistent with Figure 12. The only difference is that the gain of compensator K_{h_sq}for v_{sq}^{*}should be increased to adapt to the greater disturbances in the more important secondary sub-generator, and avoid injecting external interferences from the compensators. Thus, the compensation gains K_{h_pq}and K_{h_sq}are set as 1 and 0.35 m respectively.It should be noted that the test results in Figure 13, under a significant third harmonic flux, are independent to the MPPT characteristics. The nominal Φ_{3}related to the characteristics in Figure 3 is equal to 0.082 Wb. That is, Figure 12 shows a working point with ν_{tides}= 2.055 m/s in the MPPT characteristics. Figure 13 is performed to test the fault-tolerant control behaviors of the proposed FTC method under a condition of a greater trapezoidal deformation for the back-EMFs by increasing the flux parameter Φ_{3}in the numerical model and control strategy. The simulation results show that the proposed method can adapt well to the five-phase PMSG with different trapezoidal back-EMFs.

## 5. Experimental Verifications

_{H}= 15.

#### 5.1. Harmonic Compensation Behaviors in Healthy Conditions

**Figure 15.**Test results and their harmonic spectra in healthy conditions: (

**a**) experimental waveforms; (

**b**) comparison of phase current spectra (e.g., in phase “a”).

#### 5.2. Fault Tolerance in Single-Phase Open Conditions

_{pq}and i

_{sq}control loops by the multiple SOGIs-based compensators, which reduce the torque ripples as well as enhance the power quality of the current. In detail, the torque ripples can be constrained into 29.54% with an approximately 31.2% reduction. The suppressed fluctuations settle during 2.23 nm according to the right-most zoomed illustration in Figure 16b. Under this condition, the THD calculated by Equation (30) of phase current becomes 41.74%.

_{pq}and i

_{sq}current control loops, respectively.

_{pq}and i

_{sq}current control loops. Obviously, as depicted by the blue curve in Figure 17a, these additional interferences can be suppressed to satisfactory levels. Specially, it can be observed that the torque will contain negligible odd-order harmonics when the even ones are under-constrained, which are introduced due to the efficiency of harmonic estimation by the mSOGIs under the real-time variations of the mechanical speed that is used for harmonic resonances. Nevertheless, the compensated process guarantees the effective suppression of predominant harmonic disturbances, which inversely enhances the fault-tolerant availability of the system under the open circuit fault. Regarding the DC component in torque, the compensators maintain the level of torque since the multiple SOGIs can separate it out in real-time.

**Figure 17.**Harmonic analysis of torque and phase current: (

**a**) spectra of electromagnetic torque; (

**b**) spectra of phase current (e.g., in phase “b”).

_{em}∙ω

_{m}), are also presented in Table 3. The copper losses will be significant as one phase is open without any actions, which is mainly due to oversized current distortions and high-order harmonic interferences, as depicted in Figure 15a and Figure 16a, respectively. The proposed method has good filtering performance and fault tolerance, resulting in almost half the energy savings from the copper losses. Performance indicators are organized and analyzed, as shown in Table 3, according to the test results. With the analyzed performance of torque ripples and THD, in consequence, the proposed method presents satisfactory performance both in the healthy and faulty conditions.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**T**

_{Park13}for trapezoidal back-EMFs (contains the 1st and 3rd). Note that in the equations in Appendix A and Appendix B, mechanical angular velocity is uniformly presented as Ω:

**T**

_{Park1}for sinusoidal back-EMFs (contains only the 1st):

Symbol | Description | Value |
---|---|---|

J_{tur} | Inertia of turbine | 1.3131 × 10^{6} kg∙m^{2} |

P_{m_rated} | Generator rated power (at 50 Hz) | 1.5 MW |

V_{dc} | DC-bus rated voltage | 1700 V |

ω_{m_rated} | Rotor-rated angular velocity (at 50 Hz) | 2.618 rad/s |

n_{pp} | Pole pair number | 120 |

Φ_{1} | Magnet flux in principle sub-generator | 2.458 Wb |

Φ_{3} | Magnet flux in secondary sub-generator | 0.2731 Wb or 0.082 Wb |

R_{s} | Generator stator resistance | 0.0081 Ω |

L_{pr} | Principle sub-generator inductances | 1.2 mH |

L_{se} | Secondary sub-generator inductances | 0.88 mH |

C_{dc} | DC-bus capacitor | 13 mF |

R_{f} | Grid-side resistance | 0.1 mΩ |

L_{f} | Grid-side inductance | 1.5 mH |

Symbol | Description | Value |
---|---|---|

P_{norminal} | Nominal power of generator (at 110 Hz) | 3.3 kW |

B_{M_dc} | Friction coefficient of DC motor’s shaft | 0.091 |

B_{gen} | Friction coefficient of generator’s shaft | 0.123 |

J_{M_dc} | Inertia of DC motor | 0.02 kg·m^{2} |

J_{gen} | Inertia of generator | 0.00137 kg·m^{2} |

ω_{m_norminal} | Nominal angular velocity of generator | 230.3835 rad/s |

p | Number of pole pairs | 3 |

Φ_{1}, Φ_{3} | Magnet fluxes | 0.150 Wb, 0.0149 Wb |

R_{s} | Stator resistance | 0.540 Ω |

Load | Resistive load | 242 Ω |

L_{pr}, L_{se} | Equivalent inductance | 5.1 mH, 3.2 mH |

T_{pwm} | Switching period | 0.1 ms |

K_{h_pq}, K_{h_sq} | Compensation gains | 0.55 |

## Appendix B

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**Figure 1.**Configurations of back-to-back tidal current energy conversion systems integrated five-phase PMSG: (

**a**) machine and grid side are installed offshore and transmission via submarine cables; (

**b**) machine side in offshore and transmission via submarine cables to onshore grid side; (

**c**) five-phase PMSG is installed offshore, transmission via submarine cables to onshore machine side converter (MSC) and onshore grid side.

**Figure 4.**Structures of drive train models: (

**a**) type with gearbox in small power-scale laboratory prototype (N

_{gear}> 1); (

**b**) gearless type in real power-scale simulation setup (N

_{gear}= 1).

**Figure 5.**Transformations of principal and secondary generators: (

**a**) in principal sub-space; (

**b**) in secondary sub-space.

**Figure 7.**Bode plots of transfer functions of a SOGI with various ω

_{sogi}: (

**a**) F

_{α}(s), K

_{s}=$\sqrt{2}$; (

**b**) F

_{β}(s), K

_{s}=$\sqrt{2}$.

**Figure 8.**Description of multiple SOGIs and bode plot comparisons with different structures of SOGIs: (

**a**) structure of multiple SOGIs; (

**b**) bode plots of different design of SOGIs (e.g., extract 1st harmonic).

**Figure 11.**Comparison results of three- and five-phase PMSG-based systems: (

**a**) mechanical angular velocities; (

**b**) electromagnetic torques and mechanical torque; (

**c**) phase current of three-phase PMSG-based system; (

**d**) phase current of five-phase PMSG-based system; (

**e**) powers of three-phase PMSG-based system; (

**f**) powers of five-phase PMSG-based system.

**Figure 13.**Simulation result of compensation in q-axis current control loops when Φ

_{1}/Φ

_{3}= 9: (

**a**) machine-side waveforms; (

**b**) DC bus voltage and grid-side waveforms.

**Figure 16.**Test results and their harmonic spectra in healthy conditions by introducing the compensations: (

**a**) without compensations as phase “a” is open; (

**b**) with compensations as phase “a” is open.

Symbol | Value | Symbol | Value |
---|---|---|---|

C_{1} | 0.6406 | C_{4} | 10.778 |

C_{2} | 116.664 | C_{5} | 16 |

C_{3} | 0.4 | C_{6} | 0.0053 |

**Table 2.**General description of harmonic distribution of current/back-EMFs under healthy and single-phase open conditions.

Conditions | Healthy | Single-Phase Open (e.g., Phase “a” Is Open) | |
---|---|---|---|

Frames | |||

Original Frame | phase ‘a’ | Mainly contains 1st and 3rd harmonics (Others are regarded as disturbances) | Almost equal to 0 |

phase ‘b’ | Mainly contains 1st, 3rd, 5th and 7th harmonics | ||

phase ‘c’ | |||

phase ‘d’ | |||

phase ‘e’ | |||

Park’s Frame | pd axis | Almost equal to 0 | Mainly contains 2nd, 4th, 6th, 8th harmonics |

pq axis | Almost an offset DC component | ||

sd axis | Almost equal to 0 | Mainly contains 2nd, 4th, 6th, 8th and 10th harmonics | |

sq axis | Almost an offset DC component | ||

Homopolar | Almost equal to 0 | Mainly contains 1st, 3rd, 5th and 7th harmonics |

Indicators | Copper Losses (W) | Torque Ripples (%) | THD (%) | |
---|---|---|---|---|

Working Conditions | ||||

Healthy | No compensations | 46.65 (3.40% loss) | 28.41 | 43.23 |

By compensations | 40.41 (2.99% loss) | 23.68 | 32.84 | |

Single-Phase Open | No compensations | 69.93 (5.32% loss) | 42.96 | 67.28 |

By compensations | 34.24 (2.56% loss) | 29.54 | 41.74 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Z.; Tang, T.; Houari, A.; Machmoum, M.; Benkhoris, M.F.
An FTC Design via Multiple SOGIs with Suppression of Harmonic Disturbances for Five-Phase PMSG-Based Tidal Current Applications. *J. Mar. Sci. Eng.* **2021**, *9*, 574.
https://doi.org/10.3390/jmse9060574

**AMA Style**

Liu Z, Tang T, Houari A, Machmoum M, Benkhoris MF.
An FTC Design via Multiple SOGIs with Suppression of Harmonic Disturbances for Five-Phase PMSG-Based Tidal Current Applications. *Journal of Marine Science and Engineering*. 2021; 9(6):574.
https://doi.org/10.3390/jmse9060574

**Chicago/Turabian Style**

Liu, Zhuo, Tianhao Tang, Azeddine Houari, Mohamed Machmoum, and Mohamed Fouad Benkhoris.
2021. "An FTC Design via Multiple SOGIs with Suppression of Harmonic Disturbances for Five-Phase PMSG-Based Tidal Current Applications" *Journal of Marine Science and Engineering* 9, no. 6: 574.
https://doi.org/10.3390/jmse9060574