Research on the side converter system of wind power grid based on fractional LCL filter

The power electronic converters and grid‐connected filters were important components of the permanent magnet direct drive wind power generation system whose performance directly determines the quality of wind power generation. In past modeling, analysis, and control studies, capacitive and inductive components were often treated as integral‐order components. However, the inductance and capacitance components in the actual permanent magnet direct drive wind generator were fractional‐order components, and their electrical characteristics will change with the change of order, which had an important impact on the dynamic and static characteristics of the system. In this paper, the mathematical model of the fractional‐order LCL (FOLCL) filter was derived. Through simulation, it could be seen that the FOLCL filter can avoid resonance fundamentally. At the same time, the fractional‐order PI (FOPI) controller was introduced into the machine‐side current converter, and the parameters of the FOPI controller of the outer speed loop and the inner current loop were adjusted by using the time‐domain optimization method. The results showed that the efficiency of the FOPI controller was significantly better than that of the integer‐order PI controller in realizing maximum wind energy capture. It provides theoretical support and practical application value for the stable operation of the wind power generation system.


| INTRODUCTION
The existing research and literature have shown that inductance and capacitance components in practice are essentially fractional-order components. 1,2 Its electrical characteristics will be changed with the change of order, which will have an important influence on the dynamic and static characteristics of the system. [3][4][5] To accurately describe their electrical properties, a more accurate and flexible fractional model is needed. In this paper, fractional calculus was introduced into permanent magnet direct drive wind power system, and fractional-order LCL (FOLCL) filter was designed for gridconnected filtering using fractional inductors and capacitors, and the fractional modeling of the grid-side variable current system was carried out based on FOLCL filter. Fractional-order PI (FOPI) controller was introduced and designed for permanent magnet direct drive wind turbines and grid side converters. In conclusion, the application of fractional system modeling and fractional controller design to permanent magnet direct drive wind power system in this paper had far-reaching research significance and implementation value. The power electronic converter of permanent magnet direct drive wind power generation system is mainly to realize rectification and inverter. This topology is shown in Figure 1, which enables bidirectional energy flow.
The engine side PWM rectifier was used to control the output torque of the generator and realize the maximum power operation of the machine side, which could greatly reduce the loss of the rectifier circuit and realize the unit power factor output of the generator. The net-side PWM inverter was used to stabilize the DC link voltage and realize the decoupling control of the active and reactive power on the net side. The control strategy can be changed flexibly according to different control requirements to improve the operating characteristics of the system.
2.2 | Mathematical model of each part of permanent magnet direct drive wind power system

| Wind turbine model
In the wind power generation system, the wind turbine, as an important part of energy conversion, was mainly used to convert the kinetic energy from the air flowing through the rotor surface to the mechanical energy driving the generator rotation. 6 The output power of the whole wind power generation system was determined by the amount of wind energy converted into mechanical energy, and the safety, stability, and reliable operation of the unit are also directly affected by them. According to Bates theory, the input power of the wind turbine was 7,8 In Equation (1), ρ is the air density (kg/m 3 ), S is the area swept by the wind turbine (m 2 ), and v is the wind speed (m/s).
Due to the existence of rotational kinetic energy in the wake of the wind turbine, part of the energy was lost in the transformation process, so the energy output of the wind turbine was limited. Thus, there was a calculated value of wind energy utilization in the system, which was usually called the wind energy utilization coefficient C p . The relationship between C p and wind turbine input power Pi and output power P m was defined as follows 9 : C p is not a constant value, and its value is a nonlinear function related to tip speed ratio λ and pitch angle β, which can be further expressed as a function of λ and β, namely, C p F I G U R E 1 Topology diagram of permanent magnet direct drive wind power generation system. FOLCL, fractionalorder LCL HUANG ET AL.
| 3251 (λ, β). It can be concluded that the output power P m of the wind turbine is Equation (3).
The mechanical output power P m and mechanical torque T m of the wind turbine are Equation (4). [10][11][12] Tip speed ratio λ was introduced to express the speed of the blade, λ was defined as the ratio of tip linear velocity to wind speed, 13,14 and the expression is as follows: where R is the radius of the rotor of the wind turbine, ω is the mechanical angular velocity of the wind turbine, and n is the speed of the wind turbine. In a permanent magnet direct drive wind power system, the wind turbine and generator rotate on the same axis, so the speed of the two is equal.
The function represented by C p (λ, β) was obtained as Equation (6), and the three-dimensional diagram of pitch angle, tip speed ratio, and wind energy utilization system was drawn as shown in Figure 2. 1. Pitch angle β is constant, wind energy utilization coefficient C p can be regarded as a function of tip speed ratio λ, namely C p (λ) and λ belong to a one-to-one mapping, and an optimal tip speed ratio λ opt corresponds to a maximum wind energy utilization coefficient C pmax.

15-19
2. When tip speed ratio λ is fixed, β is the smaller pitch angle, and C p is the larger wind energy utilization coefficient. When pitch angle β = 0, C p is taken as the maximum value C pmax = 0.48, and the corresponding optimal tip speed ratio λ = λ opt = 8.1. 3. According to Betz theory, there exists a Betz limit value, namely C p (λ, β) = 0.593, but in practical engineering, C p < 0.593 is inevitable, and the actual C p that the wind turbine can achieve at present is about 0.4. [20][21][22][23][24] Therefore, when the wind speed v < v n , the pitch angle β = 0, and the λ of the wind turbine is kept at the optimal tip speed ratio λ opt so that the wind energy utilization coefficient is kept at C pmax , and the wind energy maximum power point tracking (MPPT) is realized. 25,26 When the wind speed v > v n , the pitch angle is adjusted based on the MPPT, so that the output power of the generator is limited around the rated value to ensure the safe and stable operation of the unit.

| Permanent magnet synchronous generator model
Park transformation is used to convert the electrical volume in the three-phase stationary coordinate system (abc coordinate system) into the corresponding direct axis, quadrature axis, and zero axis components in the synchronous rotating coordinate system (dq coordinate system). After Park transformation of the synchronous generator, if the three-phase generator is symmetric and in steady-state operation, the zero-axis component is 0, and the ac-straight axis components are all constant, that is the direct flow. Therefore, the other three windings can replace the rotation coordinates of the same step. [27][28][29] These three windings are rotating alternating and straight axis windings, while zero axis windings are usually not considered.
The three-phase currents i a , i b , and i c are transformed into i d , i q , and i o by Park, where θ is the rotor position or electrical angle at a certain moment. [30][31][32] For voltage and flux, there is also the following relationship in Equation (8).
The stator flux equation of permanent magnet synchronous generator (PMSG) in abc coordinate system is Equation (9). L aa , L bb , and L cc are the stator self-inductance of each phase winding, L ab , L ac , L ba , L bc , L ca , and L cb are mutual inductance between stator three-phase windings.
The electromagnetic torque equation of PMSG in abc coordinate system is Equation (10), p is the pole number of the synchronous generator.
In the motion equation of PMSG is Equation (11), T L and T e , respectively, represented load torque and electromagnetic torque, J is the moment of inertia, and B is the friction coefficient.
Park transformation was carried out on the mathematical model of PMSG in abc coordinate system, and the stator voltage equation of permanent magnet synchronous motor in dq coordinate system could be obtained from the symmetry relation as follows in Equation (12).
In Equation (12), u d is the d-axis components of the stator voltage; u q is the q-axis components of the stator voltage; i d is the d-axis components of the stator current; i q is the q-axis components of the stator current; L d is the d-axis fractions of the stator inductance; L q is the q-axis fractions of the stator inductance; R is each phase resistance of the generator stator; ω e is the angular velocity of the generator and ω e = p ω ; and ψ d , ψ q is the d-axis components of the stator flux.
In the dq coordinate system, the flux equation is Equation (13).
Equations (12) and (13) are combined, then can get the electromagnetic torque Equation (14) in the dq coordinate system.

| DC link model
The main function of the DC link is to filter and stabilize DC side voltage by shunt capacitor, the model is as follows in Equation (15).
P c was the instantaneous power flowing through the current container, P dc1 was the instantaneous power output by the converter on the machine side, and P dc2 was the instantaneous power output by the DC link. On the premise of ignoring system loss, to ensure lossless transmission energy between PWM converters on the grid side of the machine, it was necessary to control the grid side converters to make P dc1 = P dc2 , P c = 0 complete. However, it was difficult to achieve the goal of keeping the DC bus voltage constant in actual operation, mainly because of the delay of the control of the grid side converter.

| Machine-side PWM converter model
The PWM converter on the engine side was mainly composed of an AC circuit, switching circuit, and DC link, and its structure is shown in Figure 3.
In Figure 3, e a , e b , e c is the induced potential of the PMSG; u q is the q-axis components of the stator voltage; u AO , u BO , u CO is the three-phase voltages of the converters on the machine side; T 1 -T 6 is IGBT; R L is resistance load; and i dc is DC bus capacitor current.
The voltage equation of the PWM converter at the machine side: The mathematical model of the PWM converter at the machine side under abc stationary coordinate system is Equation (17).
After Park transformation, the mathematical model of the machine-side PWM converter in the dq coordinate system is as follows in Equation (18).

| Mathematical model and characteristic analysis of FOLCL filter
The block diagram of the transfer function of the FOLCL filter in the frequency domain is shown in Figure 4.
The grid-side current of the FOLCL filter can be expressed as follows.
Equation (19) can be expressed as a single-input single-output system, the transfer function G 1 (s) of the FOLCL filter is Equation (20).
The circuit diagram of the machine-side PWM converter of the permanent magnet direct drive wind power generation system.
Its frequency domain expression is Equation (21).
The resonant frequency of the FOLCL filter can be obtained as follows: From Equations (20)- (22), we can draw the following conclusions: 1. When α = β = 1, the resonance frequency expression of the FOLCL filter is the same as that of the integer-order LCL (IOLCL) filter. Therefore, the IOLCL filter is a special case of the FOLCL filter. 2. Equation (22) can be considered the design formula of the FOLCL filter. Compared with the IOLCL filter, it has two more adjustable design parameters, when α + β ≠ 2, the denominator of ｜G 1 (jω)｜ corresponding to ω is not zero, and the resonance peak did not exist, so the FOLCL filter can effectively avoid the resonance. 3. When α and β were further away from 1, the frequency response of the FOLCL filter was smoother. At this time, the power loss brought by the FOLCL filter was larger. For the FOLCL filter, the introduction of fractional inductance and capacitance made the FOLCL filter more flexible, and could effectively avoid resonance fundamentally, without additional damping.
However, it will inevitably bring power loss, so the order determination of fractional components needs to be considered comprehensively between smooth frequency response and control loss.

| Simulation of filtering characteristics of FOLCL filter
Based on the MATLAB platform, the frequency characteristic curve of the FOLCL filter was drawn. The simulation was divided into two groups: one group had fixed fractional inductance order α and changed the fractional capacitance order β, and one set had fixed fractional capacitance order and changed the fractional inductance order α.
The frequency characteristics of FOLCL filters with fractional inductor and capacitive component order changes are shown in Figure 5.
Comprehensive Figure 5A,B analysis showed that in the logarithm amplitude-frequency characteristics, the low-frequency attenuation rate is −20αdB/dec and the attenuation rate of high frequency is −(40α + 20β)dB/ dec. In the phase-frequency characteristics, the decay rate in the low-frequency band was −πα/2, and the decay rate in the high-frequency band was −(α + β/2)π. When the IOLCL filter was used in the network side converter system, the amplitude and frequency characteristics of the resonant peak appear. However, when the FOLCL filter was used, the inductance and capacitance order were not 1, there was no resonance spike so the system is always in a stable state.

| Fractional modeling and control of network side converter system
The main functions of the netside converter are as follows: 1. To stabilize the DC bus voltage, so as to achieve the purpose of stable operation of the system; 2. The side converter output DC inverter to alternating current; 3. Alternating current that ensures synchronous output grid frequency, phase sequence, phase, and waveform; 4. Reduce the current harmonics incorporated into the power grid. 3.3.1 | Fractional control structure of the netside converter system Figure 6 shows the topology of the network side converter system based on the FOLCL filter, where three-phase grid voltages are e a , e b , and e c , gridconnected currents are i a , i b , and i c , converter bridge arm output voltage are v a , v b , and v c , and FOLCL filter inductance and capacitance order are α and β, and L T = L i + L g . The filter resistors at the converter side and the grid side are, respectively, R i and R g , and let The mathematical model of the grid-side converter in abc coordinate system is Equation (23).
Its matrix form is Equation (24).
By applying fractional dq transformation to Equation (23), the mathematical model in the twophase rotating coordinate system can be obtained as follows: The active and reactive power transmitted to the power grid by the grid-side converter system is Equation (25), the grid voltage vector direction was fixed on axis d, and the grid voltage projection on axis q was zero, which was the vector control based on grid voltage orientation.
According to Equation (26), active power and reactive power can be controlled by adjusting the current components of the d and q axes.
The feed-forward control of grid voltage is usually used to compensate for the influence of grid voltage variation on system control. Thus, the current inner loop decoupling control scheme as shown in Figure 7 can be obtained.
In this way, the structure diagram of the double closed-loop control system based on grid voltage orientation can be obtained as shown in Figure 8.
In addition, because the resistance value of R i and R g is small in the analysis and calculation, the impact on the system is usually ignored and the resistance value is directly set to zero.

| Set the parameters of the FOPI controller of the inner loop of the grid-side current
The general form of the FOPI controller is Equation (27).
When SVPWM modulation is adopted and the sampling delay and the small inertia characteristic of PWM control are ignored, the fractional controlled object of the inner loop of the grid-side current can be simplified as follows.
In this way, the robustness of the phase margin and gain variation of controlled objects were taken as design indicators to meet the following three design principles.
The step response curves of the two groups of transfer functions controlled by FOPI and integerorder PI (IOPI) for the fractional controlled object under the inner loop of the grid-side current were shown in Figure 9.
As could be seen from the figure, the overshoot in step response under the IOPI controller was 3.5% and the adjustment time was 0.00144 s, while the overshoot in step response under the FOPI controller was 1.2%. The results showed that the performance of the FOPI controller had been improved.
The voltage sampling delay T V in the outer voltage loop control and the equivalent hourly constant 3T i in the inner current loop were combined to obtain the structure diagram of the outer voltage loop control as shown in Figure 10.
The step responses of the FOPI controller and IOPI controller were simulated and compared as shown in Figure 11.
As could be seen from Figure 11, the overshoot in step response under the IOPI controller was 46.8% and the adjustment time was 0.0049 s, while the overshoot in step response under the FOPI controller was 23.4% and the adjustment time was 0.0036 s. The results showed that the control performance of the FOPI controller had obvious advantages; it had better dynamic response performance.

| Technical requirements of machine-side converter system
In permanent magnet direct drive wind power system, the main functions of the converters on the side of the machine include: 1. Controlling the speed of the synchronous generator to realize the maximum power tracking of wind energy; 2. Monitor the working state of each component of the side converter system to ensure the safe and stable operation of the system; 3. The output voltage and frequency of the PMSG are unstable alternating current converted to constant direct current.
The machine-side current converter system is a kind of multi-variable, strongly coupled, and nonlinear system. Under the condition of large internal and external disturbances or the change of grid side load, the FOPI controller has superior performance in speed, flexibility and robustness, and so forth. It can realize the DC voltage stabilization and the maximum power tracking of wind energy better and has better system running stability performance.

| Maximum power tracing policy
In any wind speed within a certain range, the MPPT strategy of variable speed wind turbine was preferred to make the wind turbine capture the corresponding Maximum wind energy under the wind speed.
In this paper, the optimal tip speed ratio control was adopted to ensure that the ratio was maintained in the best state to achieve the maximum power tracking of Step response of current inner loop controlled by fractional-order PI (FOPI) and integer-order PI (IOPI) in Grid-side system.
F I G U R E 10 Simplifified voltage outer loop fractional-order control structure block diagram.

F I G U R E 11
Voltage outer loop step response curve under fractional-order PI (FOPI) and integer-order PI (IOPI) control. wind energy by controlling the ratio of the generator speed to wind speed.

| Vector control of machine-side converter system
Zero d shaft current vector control strategy had been adopted to control the machine-side converter system in this paper. In general, using the synchronous rotating coordinate system will be equivalent to DC motors, PMSG to achieve separate adjust the d-and q-axis components of stator current, torque, and synchronous generator and excitation to decoupling control. Eventually, machine-side converter control performance of the system had been greatly improved.
The double closed-loop control structure diagram of the rotational speed outer loop and current inner loop of the machine side current converter system is shown in Figure 12.

| Design of FOPI controller for machine-side converter system
To capture maximum wind energy, first, the optimal speed value of the generator rotor corresponding to the wind turbine at the wind speed v was determined according to the optimal tip speed ratio control method and was set as the given speed ω* of the speed outer ring. Then, the actual speed ω and the given speed ω* are compared, and the difference Δω was passed through the FOPI controller to obtain the given value of active current i* q . At this stage, the given value of reactive current i* = 0 d was always maintained. Finally, in the current inner loop, the actual voltage feeding quantity u* d and u* q of the generator were the sum of the feedforward compensation term and the difference between the actual current value i d and i q and the given current value i* d and i* q after adjustment by the FOPI controller, the control schematic diagram of the permanent magnet direct drive wind power system is given in Figure 13 The PMSG used in this paper is a hidden pole type, so the armature inductance of the ac direct axis is equal, that is L d = L q = L. Since the control object parameters of active and reactive current loops are the same, they can adopt the same set of FOPI control parameters. Specific control rules are as follows: The time-domain optimization method was used to design the parameters of the current inner loop FOPI controller on the side of the machine:"integral error (ITAE)" was taken as the optimization performance index, and the genetic algorithm was used to search the initial value and set it as the IOPI control parameter to obtain the FOPI controller parameters as shown in Table 1.
In Table 1, the step response overshoot of the IOPI controller was 4.3% and the adjustment time was F I G U R E 12 Fractional-order double closed-loop control block diagram of a machine-side converter.
F I G U R E 13 Fractional-order control structure block diagram of generator-side converter system of permanent magnet direct drive wind power generation system. 0.00199 s, and the step response overshoot of the FOPI controller was 1.7%, the results showed that the performance of the FOPI controller was improved, the step response curves of the two are shown in Figure 14. 4.4.2 | Setting parameters of FOPI controller of the external ring of machine side speed After comparing the given reference speed ω* with the actual generator speed ω, the reference value of active current i* q could be obtained by the FOPI controller. Since the zero d-axis current vector control strategy was adopted, the given reference value of reactive current i* = 0 The time-domain optimization method was used to design the parameters of the external loop FOPI controller on the engine side, "ITAE" was taken as the optimization performance index, and the genetic algorithm was used to search the initial value and set it as the IOPI control parameter to obtain the FOPI controller parameters as shown in Table 2.
The step responses of the FOPI controller and the IOPI controller on the machine side are compared. In Table 2, the overshoot in step response under the IOPI controller was 37.6% and the adjustment time was 0.00515 s, and the overshoot in step response under the FOPI controller was 5.9% and the adjustment time was 0.00417 s. The results showed that the control performance of the FOPI controller had obvious advantages, the result is shown in Figure 15 The simulation model of the turbine side converter system was composed of the wind turbine model, PMSG model, turbine side converter model, DC side, and turbine side converter control module (Generatorcontrol). Specific simulation model parameters of the machine-side system are shown in Table 3. According to the simulation results, the FOPI controller designed above for the speed outer loop and current inner loop was simulated and verified.

| Wind turbine model building and simulation
The control of the machine side converter is actually accomplished by controlling the PMSG, which adopts the double closed-loop control structure of the outer speed loop and the inner current loop. Figure 16 shows the internal structure simulation model of the Generatorcontrol module. The generator speed was adjusted by the speed ring FOPI controller of the q-axis and then flowed through the current ring FOPI controller to get the control voltage of the q-axis. The d-axis control was another current loop using zero d-axis current control; the simulation model of side-converter control is shown in Figure 16. FOPI and IOPI control modes were simulated and compared for the permanent magnet direct drive wind power generation system. The blade tip speed ratio simulation diagram and wind energy utilization coefficient simulation are shown in Figures 17 and 18.
The simulation results showed that the optimal tip speed ratio control method could make the tip speed ratio and wind energy utilization coefficient synchronously follow the change of wind speed and fast track to the best value, that was, the maximum power tracking of permanent magnet direct drive wind power generation system was realized.
In fact, the control of the converter on the side of the machine was accomplished by controlling the PMSG, which adopted the double closed-loop control structure of the outer speed loop and the inner current loop. The simulation waveforms of the speed of the PMSG and the stator current of the d and q axes are shown in Figures 19  and 20.
From Figure 19, compared with the IOPI controller, the dynamic performance of the speed regulating system using the FOPI controller was more advantageous. When the wind speed increases from 11 to 14 m/s at 0.25 s, the speed of the IOPI control system rises to 123.7 r/min at 0.2524 s, while the speed of the FOPI control system rises to 120.4 r/min at 0.2522 s. Moreover, the FOPI control system could quickly follow the optimal speed of 108.29 r/min. It had been indicated that the FOPI control system had better start-up performance, stronger antiinterference ability, smaller overshoot, and faster response time than the IOPI control system. From Figure 20, when the wind speed changes, the q-axis current component of the generator stator changes with it, while the d-axis current component was always maintained near the value of 0. In this way, the control requirements of the system were realized, and the correctness of the proposed control strategy based on the zero d-axis current was verified.

| CONCLUSION
By introducing fractional-order calculus theory and designing the FOLCL filter, fractional-order modeling and FOPI control are carried out on the grid side of the permanent magnet direct drive wind power system. Then, the optimal tip speed ratio control method in the MPPT strategy was used to achieve the maximum wind energy capture of wind turbines, and then the vector control strategy based on rotor field orientation was used to control the side converter, which further realized the

F I G U R E 15
Step response of speed outer loop under fractional-order PI (FOPI) and integer-order PI (IOPI) control of machine side system.
T A B L E 3 Machine-side system simulation parameters.

Parameter name Value
Rotor moment of inertia, J (kg m 2 ) 5 Pitch angle (β) 0 control of permanent magnet wind synchronous generator. At the same time, the double closed-loop control structure of the FOPI controller was designed based on the control strategy, and the simulation model of the machine-side system based on IOPI and FOPI controller was established in the MATLAB environment. The reliability of fractional modeling and control and the advantages of improving the generation efficiency and grid-connected power quality of permanent magnet direct drive wind power generation system were proved by analyzing the experimental results of blade tip speed ratio, wind energy utilization coefficient, synchronous F I G U R E 16 Machine side converter control simulation model.

F I G U R E 17
Comparison chart of tip speed ratio between integer-order PI (IOPI) control and fractional-order PI (FOPI) control.
F I G U R E 18 Comparison chart of wind energy utilization coefficient under integer-order PI (IOPI) control and fractionalorder PI (FOPI) control.
generator speed, and the current of d-axis and q-axis on the side of the generator.