energies
Article Predictive Direct Flux Control—A New Control Method of Voltage Source Inverters in Distributed Generation Applications
Jiefeng Hu
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China; [email protected]; Tel.: +852-2766-6140
Academic Editor: Frede Blaabjerg Received: 16 February 2017; Accepted: 21 March 2017; Published: 23 March 2017
Abstract: Voltage source inverters (VSIs) have been widely utilized in electric drives and distributed generations (DGs), where electromagnetic torque, currents and voltages are usually the control objectives. The inverter flux, defined as the integral of the inverter voltage, however, is seldom studied. Although a conventional flux control approach has been developed, it presents major drawbacks of large flux ripples, leading to distorted inverter output currents and large power ripples. This paper proposes a new control strategy of VSIs by controlling the inverter flux. To improve the system’s steady-state and transient performance, a predictive control scheme is adopted. The flux amplitude and flux angle can be well regulated by choosing the optimum inverter control action according to formulated selection criteria. Hence, the inverter flux can be controlled to have a specified magnitude and a specified position relative to the grid flux with less ripples. This results in a satisfactory line current performance with a fast transient response. The proposed predictive direct flux control (PDFC) method is tested in a 3 MW high-power grid-connected VSI system in the MATLAB/Simulink environment, and the results demonstrate its effectiveness.
Keywords: voltage source inverter (VSI); predictive control; inverter flux
1. Introduction Pulse width modulation (PWM) voltage source inverters (VSIs) have been widely used in power electronics, and their control has been extensively studied in the last decades. One of the most popular applications of VSIs is electric drives for speed control and electromagnetic torque regulation [1]. Recently, inverter-interfaced distributed generation (DG) systems (wind turbines, photovoltaic (PV), wave generators, etc.) have raised concern. These DGs are either connected to a common bus to form a small isolated power system, or connected to the main utility grid. The power supplied by the DGs as well as the DG output voltage and current can be flexibly regulated by controlling VSIs in various operation conditions [2,3]. In the VSI applications mentioned above, generally two individual quantities are controlled in closed-loop strategies for inverters by means of switching. Traditionally the d- and q-axis current components are directly controlled in electric drives [4]. Later on, direct torque control (DTC) was proposed and it eliminates the need for a current regulator and provides direct control of the motor electromagnetic torque [5]. Derived from DTC, direct power control (DPC) was proposed to control rectifiers and inverters by using the active and reactive powers as control variables in DGs [6]. Similarly, direct current control and direct voltage control were proposed for grid-connected or isolated VSI systems [7,8]. However, the torques (or active and reactive powers, currents and voltages) are not the only quantities that can be regulated directly by the inverter switching. The control of the inverter flux has
Energies 2017, 10, 428; doi:10.3390/en10040428 www.mdpi.com/journal/energies Energies 2017, 10, 428 2 of 11 raised concern in the last few years because the inverter flux is less sensitive to external disturbance for its voltage integral feature [9–11]. Inverter flux control was first proposed by Chandorkar et al. [9,10] for inverter stand-alone operations. Nevertheless, it has not been applied in grid-connected inverters. Inverter flux was adopted in grid-connected converters [11]. However, the inverter flux was just used to estimate the active and reactive powers in order to eliminate the instantaneous errors during measurement. In other words, the direct control variables of the inverters are active and reactive powers rather than the inverter flux. To the best of the authors’ knowledge, there are no existing papers working on direct flux control for grid-connected VSI systems. In addition, the magnitude of the inverter flux is related to the inverter voltage while the position of the inverter flux links to the voltage frequency. Therefore, inverter flux control could be a promising alternative for grid-connected renewable energy systems to performance grid support. Unfortunately, the existing inverter flux control strategies are still under development. The only approach reported in the literature is the switching table–based direct flux control, or SDFC [9,10]. In this scheme, the inverter switching is selected from a predefined switching table according to the position of the inverter flux. As a result, similar to conventional DTC and DPC, large flux ripples are the main drawback of SDFC. This leads to deteriorated power and current quality. In order to address the disadvantages of SDFC, a new inverter control method is proposed in this paper by adopting predictive control to the inverter flux concept. The controller uses the system model and all the possible switching states to predict the behavior of the inverter flux, namely the flux amplitude and the flux position, at the next sampling instant. Then, a cost function is employed as a criterion to evaluate the inverter flux performance. Finally, the voltage vector or the switching state that minimizes the cost function will be applied during the next sampling interval. In this way, the inverter flux can be well controlled to track the reference. Also, it is necessary to point out that the proposed method in this paper falls into the category of model predictive control (MPC), while the method presented in [11] belongs to the category of vector sequence–based predictive control. The cost function developed here presents totally different meanings to the cost function described in [11]. In our work, the optimal voltage vector is selected according to the cost function, whereas the voltage vector sequences are selected according to a predefined switching table. The cost function is used to calculate the applied duration of the selected voltage vectors. The rest of this paper is organized as follows. The VSI system configuration and the inverter flux concept are described in Section2. The conventional SDFC method is presented in Section3, while the new predictive direct flux control (PDFC) is introduced in Section4. The effectiveness of the proposed method is verified in Section5. The conclusion is presented in Section6.
2. Flux Concept in VSI Systems The power circuit considered is shown in Figure1. The energy resources are interfaced with the grid through the three-phase two-level VSI with six insulated gate bipolar transistors (IGBTs). The DC output from the energy resources could be from the solar PV panels, the permanent magnet synchronization wind generator with diode rectifiers, etc. L and R represent the line inductor and resistor, respectively. Depending on the switching states, the inverter can be controlled to generate eight possible output voltages including six active vectors and two null vectors. They can be expressed in complex space vectors as:
( 2 j(i−1) π V e 3 (i = 1 ··· 6) V = 3 dc (1) 0 (i = 0, 7) where Vdc is the dc-link voltage. Figure2 shows the inverter output voltages in terms of amplitude and position in the stationary α-β frame. For instance, if the upper switches of phases A and B are turned ON while the upper switch of phase C is turned OFF, voltage vector V2 with an amplitude of ◦ 2/3 Vdc and a phase angle of 60 will be generated. It is noted that this two-dimensional plane can be divided into six sections, which will be explained and used in inverter flux control. The mathematical equations of the system equivalent circuit can be described as: Energies 2017, 10, 428 3 of 11 Energies 2017,, 10,, 428428 3 of 11
dI V ==+RI + LdI ++ E (2) VI=+RLdt + E (2) dtt 3 33∗ 3 P = ==+Re33{EI }* = (()Eα Iα + Eβ Iβ) (3) PEIEI2 Re{EI } 2 ()αα ββ (3) 22 3 ∗ 3 Q = Im33{EI } = (Eβ Iα − Eα Iβ) (4) 2==−33* 2 QEIEIIm{EI } ()βα αβ (4) where E and I are the grid voltage vector and22 the line current vector, respectively. R is the line resistance, andwhereL is theE andand line I inductance. are the gridP voltageand Q arevector the and active the and line reactive current power vector, flow respectively. between theR isis system thethe lineline and theresistance, utility grid. and Similar L isis thethe to lineline the inductance.inductance. flux definition P andand in Q an areare electrical thethe activeactive machine, andand reactivereactive the inverter powerpower flowflow flux between vectorbetween and thethe the gridsystem flux vectorand the can utility be defined grid. Similar as [10 ]:to the flux definition in an electrical machine, the inverter flux t vector and the grid flux vector can be defined as [10]:Z ψ τ V = tt Vd (5) =τ ψV −∞Vd (5) −∞ t Ztt ψ ==τEdτ (6) ψEE Ed (6) −∞ −∞ ItIt is is noted noted that that the the inverter inverter flux flux andand thethe gridgrid flux flux are are the the rotating rotating space space vectors vectors with with a specified a specified amplitude and angular speed. Here, the angle between ΨV and ΨE, indicated as the power angle, is amplitudeamplitude and and angular angular speed.speed. Here, the the angle angle between between Ψψ andV and Ψ ψ, indicatedE, indicated as the as power the power angle, angle, is is defineddefined as: as: δ δ=δ−δ= δ − δ p pp VVEE (7) (7) wherewhereδV δandV andandδ EδEare areare the thetheangles anglesangles ofofof ψΨVV andand ΨψEE,, , respectively.respectively. respectively.
AB C DC DC L R output Vdc from dc Grid C energy resources
FigureFigure 1. 1.Equivalent EquivalentEquivalentcircuit circuitcircuit of ofof thethethe grid-connectedgrid-connected voltagevoltage voltage sourcesource source inverterinverter inverter (VSI)(VSI) (VSI) system.system. system.
ImIm
V33(010)(010) V22(110)(110)
V44(011)(011) V11(100)(100) Re V00,, V77
V55(001)(001) V66(101)(101)
FigureFigure 2. 2. Possible PossiblePossible voltagevoltage vectorsvectors generatedgenerated generated byby by thethe the inverter.inverter. inverter.
3.3. Conventional Conventional Switching Switching Table–Based Table–Based DirectDirect FluxFlux Control (SDFC) (SDFC) ControlControl of of the the flux flux vectorvector has been been found found to to sh showow a good a good dynamic dynamic response response and robustness. and robustness. In conventional SDFC, two variables that are controlled directly by the inverter are |ΨV| and δpp.. InIn otherother In conventional SDFC, two variables that are controlled directly by the inverter are |ψV| and δp. words, the vector ΨV isis controlledcontrolled toto havehave aa specifiedspecified magnitudemagnitude andand aa specifiedspecified positionposition relativerelative toto In other words, the vector ψV is controlled to have a specified magnitude and a specified position the vector ΨE,, asas shownshown inin FigureFigure 3.3. TheThe controlcontrol strategystrategy of the SDFC is depicted in Figure 4. The relative to the vector ψE, as shown in Figure3. The control strategy of the SDFC is depicted in Figure4.
Energies 2017, 10, 428 4 of 11
Energies 2017, 10, 428 4 of 11
The signalsEnergies 2017dF ,and 10, 428d A are first generated by two hysteresis comparators according to the tracking4 of 11 errors betweensignals the dF estimatedand dA are andfirst referencedgenerated by values two hysteresis of the flux comp andarators power according angle. The to the voltage tracking vector errors is then selectedbetweensignals from d theF and a look-upestimated dA are first table and generated (Tablereferenced1) accordingby values two hysteresis of to thed Fflux, dcompA andandarators powerδV. Similar accordingangle. The to theto voltage the switching tracking vector table–basederrors is then DTCselectedbetween and DPC, from the theestimated a look-up SDFC strategyand table referenced (Table is based 1) valuesaccording on theof the factto dfluxF that, d Aand and the power δ effectsV. Similar angle. of eachto The the inverter voltageswitching vector voltage table–based is then vector on DTCselected and from DPC, a thelook-up SDFC table strategy (Table is 1) based according on the to fact dF, d thatA and the δV .effects Similar of to each the switchinginverter voltage table–based vector |ψV| and δ are different. This is summarized in Table1[ 9], where Sk is the sector number in the α-β onDTC |Ψ andV| and DPC, δ are the different.SDFC strategy This isis summarbased onized the infact Table that 1the [9], effects where of S eachk is the inverter sector voltage number vector in the plane given by the position of ψV, being dF = 1 if |ψV|* > |ψV|, dF = 0 if |ψV|* < |ψV|; and dA = 1 αon-β |planeΨV| and given δ are by different.the position This of is Ψ summarV, beingized dF = in1 if Table |ΨV |*1 [9],> | ΨwhereV|, dF S=k 0 is if the |Ψ sectorV|* < | numberΨV|; and in d theA = 1 if δ* > δ, dA = 0 if δ* < δ. ifα -δβ* plane> δ, dA given = 0 if byδ*
Figure 3. Illustration of inverter flux and grid flux in the α-β plane. FigureFigure 3. 3.Illustration Illustration ofof inverterinverter flux flux and and grid grid flux flux in inthe the α-βα plane.-β plane.
Figure 4. Conventional directdirect fluxflux co controlntrol strategy strategy of of inverters. inverters. Figure 4. Conventional direct flux control strategy of inverters.
Table 1. Switching table–basedtable–based directdirect flux flux control control (SDFC) (SDFC) switching switching table. table. Table 1. Switching table–based direct flux control (SDFC) switching table. Sector Number (Location of ΨV) Sector Number (Location of ΨV) Flux Regulation S1S1Sector S2S2 NumberS3S3 S4S4 (Location S5S5 S6S6 of ψV) Flux Regulation dF = 1, increase ||ΨΨVV|| S1VV22 VV S233 VV4 4 S3 VV5 5 S4VV6 6 V S5V1 1 S6
dF = 0, decrease ||ΨΨVV|| VV33 VV44 VV5 5 VV6 6 VV1 1 VV2 2 dF = 1, increase |ψV| V2 V3 V4 V5 V6 V1 dF = 0, decreasedA == 0,0, | ψ nullnullV| vectorvectorV is3is appliedappliedV4 to to decrease Vdecrease5 V δ 6δp p V1 V2 dA = 0, null vector is applied to decrease δp 4.4. PredictivePredictive DirectDirect Flux Control 4. PredictiveAsAs explainedexplained Direct Flux before, Control in SDFC, the positionposition ofof ththee inverterinverter flux flux is is identified identified according according to to an an α -αβ- β plane divided into six sectors. There is one voltage vector and only that one selected within a whole planeAs explained divided into before, six sectors. in SDFC, There the is positionone voltage of vector the inverter and only flux that is one identified selected accordingwithin a whole to an α-β sector.sector. InIn fact,fact, thethe vector selected according toto ththee switchingswitching table table is is not not necessarily necessarily the the best best one one for for plane divided into six sectors. There is one voltage vector and only that one selected within a whole sector. In fact, the vector selected according to the switching table is not necessarily the best one for Energies 2017, 10, 428 5 of 11
controllingEnergies 2017 the, 10 inverter, 428 flux amplitude and power angle, especially when the inverter flux position5 of 11 is located near the edge of the sectors. Therefore, it is expected that the voltage vectors are always chosen accordingcontrolling to some the inverter specified flux criteria amplitude regardless and power of the angle, inverter especially flux position.when the inverter flux position isBased located on near the abovethe edge discussion, of the sectors. a PDFC Therefore, is proposed it is expected for VSIs that in thisthe voltage paper. Invectors fact, severalare always kinds of controlchosen schemes according have to been some developed specified criteria under theregardless name of of predictive the inverter control, flux position. such as deadbeat predictive Based on the above discussion, a PDFC is proposed for VSIs in this paper. In fact, several kinds control, hysteresis-based predictive control, trajectory-based predictive control, MPC, etc. Due to its of control schemes have been developed under the name of predictive control, such as deadbeat flexible control scheme that allows the easy inclusion of system constraints and nonlinearities of the predictive control, hysteresis-based predictive control, trajectory-based predictive control, MPC, etc. system,Due MPC to its has flexible raised control much concernscheme inthat the allows last few the years easy [12 inclusion,13]. In MPC,of system different constraints definitions and of the cost functionnonlinearities are possible,of the system, considering MPC has different raised much norms concern and including in the last several few years variables [12,13]. and In MPC, weighting factorsdifferent [14–16 definitions]. Taking theof the advantages cost function of MPC, are po a newssible, PDFC considering strategy different is developed. norms and Different including from the use ofseveral hysteresis variables comparators and weighting and factors the switching [14–16]. Taki tableng inthe SDFC, advantages the selection of MPC, a of new voltage PDFC vectors strategy of the proposedis developed. PDFC is Different based on from evaluating the use of a hysteresis defined costcomparators function. and the switching table in SDFC, the selectionGenerally, of voltage the proposed vectors of controllerthe proposed consists PDFC is based of three on evaluating main steps, a defined namely cost function. flux prediction, optimal voltageGenerally, vector the proposed selection controller and weighting consists factorof three tuning, main steps, as illustrated namely flux in Figureprediction,5. optimal voltage vector selection and weighting factor tuning, as illustrated in Figure 5.
E
Vdc V αβ Flux · Prediction V · |ψV|(k+1) δp(k+1) * Optimal Vector |ψV| Selection * δp PWM
Figure 5. Proposed predictive direct flux control (PDFC) scheme. Figure 5. Proposed predictive direct flux control (PDFC) scheme. Step 1: Flux prediction Step 1: Flux prediction The system variables that need to be predicted are the inverter flux amplitude |ΨV| and the k +1 k p ψ ψ s s powerThe system angle δ variables. According that to need Equation to be (5), predicted we have areV the = inverterV + VT flux, where amplitude T is the |samplingψV| and the period and V is the possible inverter voltage vectors.k+ As1 explainedk before, V can be controlled to power angle δp. According to Equation (5), we have ψV = ψV + VTs, where Ts is the sampling period and Veightis the space possible voltage inverter vectors, voltage depending vectors. on the As explainedswitching states. before, TheV caninverter be controlled flux vector to can eight be space voltagepredicted vectors, using depending α and β components on the switching as: states. The inverter flux vector can be predicted using α kk+1 =+ and β components as: ψψVVαααVT s (8) ψk+1 = ψk + V T Vα kk+1 =+Vα α s (8) ψψVVβββVT s (9) k+1 k ψ = ψ β + VβTs (9) Subsequently, the inverter flux amplitudeVβ at theV next (k + 1)th sampling instant can be expressed as:Subsequently, the inverter flux amplitude at the next (k + 1)th sampling instant can be expressed as:
kkk+++11212r ψ =+(ψ αβ)(2 ψ ) 2 (10) ψk+1 VVV= (ψk+1) + (ψk+1) V Vα Vβ (10) For the inverter flux angle, it can be predicted by: For the inverter flux angle, it can be predicted by: ψk +1 δ=k +1 V α + V arctan(+ ) k 1 (11) k+1 ψk 1ψVα δ = V(β ) V arctan k+1 (11) ψVβ Since the grid is a stiff AC power system, the grid voltage vector rotates at a constant angular speed.Since theTherefore, grid is the a stiffgrid ACflux powerangle can system, be simply the predicted grid voltage as: vector rotates at a constant angular speed. Therefore, the grid flux angle can beδ=δ+ω⋅ simplykk+1 predicted as: E EsT (12) δk+1 = δk + ω · T (12) where ω is the angular speed of the grid voltageE inE radians sper second, i.e., the angular speed of the wheregridω flux.is the Now angular that the speed future of states the grid of the voltage inverter in flux radians angle per and second, the grid i.e.,flux theangle angular have been speed of the gridobtained, flux. the Now power that angle the future can thus states be calculated of the inverter as: flux angle and the grid flux angle have been obtained, the power angle can thus be calculated as: Energies 2017, 10, 428 6 of 11
k+1 k+1 k+1 δp = δV − δE (13)
Substitute Equations (12) and (13) into Equation (8), and the vector that produces the minimum J will be selected to control |ψV| and δp. Step 2: Optimal inverter voltage vector selection After the future behavior of the system is predicted, the next step is to design a cost function (or selection criteria) to evaluate all the possible switching states. Actually, different formulations of the cost function are possible, depending on which variables need to be controlled [15]. In this paper, the cost function is chosen in such a way that both the flux and angle are as close as possible to the referenced values, which is defined as: r 2 2 = ( ψ ∗ − ψk+1 ) + (δ∗ − δk+1) Jmin k1 | V | V k2 p p (14)
∗ where k1 and k2 are the weighting factors, |ψV|* and δp are the referenced inverter flux amplitude and power angle, respectively. In this paper, weighting factors are obtained with the purpose of getting a trade-off between the flux and power angle. As there are eight possible voltage vectors, there will be eight possible values of the cost function. Next, the voltage vector that can minimize the cost function will be selected to control the inverter. Step 3: Weighting factor determination In grid-connected inverters where active power and reactive power are the control objectives, the weighting factors of the cost function can be decided in a simple way by equalizing k1 and k2, because the active power and reactive power show the same control priority and they have the same order of magnitude. However, the control objectives in this paper are the flux magnitude |ψV| and the power angle δp. The first term in Equation (14) reduces the flux ripples, while the second term contributes to power angle ripple reduction. For example, a larger value for k1 implies a higher priority for the flux regulation. Now it can be seen that the unit of |ψV| is Webers while the unit of the δp is radians, and they show different orders of magnitude. Therefore, it is necessary to design a suitable weighting factor determination approach in this case. Here, a two-step tuning method is developed. In coarse tuning, the preliminary ratio of k1/k2 can be obtained according to the nominal values of the power angle and the flux. For instance, if the nominal power angle is 0.7 radians and the nominal flux amplitude is 12 Wb for a grid-connected inverter, then k1:k2 can be set as 0.7:12. In this way, both the flux and the power angle can be controlled in a balance. After that, k1 and k2 will be adjusted slightly in fine tuning until a satisfactory performance is obtained. Actually there is some specified research focused on weighting factor optimization [17,18]. However, the topic of weighting factor optimization is out of the scope of this paper.
5. Results and Discussion The proposed PDFC of inverters is tested in a 3 MW high-power VSI system connected to 3.3 kV grid in a MATLAB Simulink (MATLAB 2016a, www.mathworks.com) environment. The power circuit of the system is shown in Figure1. The parameters of the VSI system are listed in Table2. To demonstrate the effectiveness of the proposed PDFC scheme for VSIs, the conventional SDFC will be employed for comparison. For SDFC, the band width of the hysteresis comparators of the flux magnitude and power angle are HF = 0.075 and HA = 0.01. For PDFC, the weighting factors are k1 = 1, k2 = 18. The sampling rate for both strategies is 10 kHz, leading to a 2.02 kHz average switching frequency for SDFC and a 1.95 kHz average switching frequency for PDFC. Energies 2017, 10, 428 7 of 11
Table 2. System parameters.
Energies 2017, 10, 428 7 of 11 Quantity Symbol Value Line ResistanceTable 2. System parameters.R 0.51 Ω Line inductance L 20 mH Quantity Symbol Value Line-line grid voltage E 3.3 kV (rms) LineSystem Resistance frequency f R 50 Hz0.51 Ω Line inductance L 20 mH DC-link voltage Vdc 10 kV Line-lineSampling grid voltage period Ts E 1003.3µs kV (rms) System frequency f 50 Hz DC-link voltage Vdc 10 kV 5.1. Steady-State Performance Sampling period Ts 100 μs
The5.1. system Steady-State steady-state Performance performance is shown in Figure6. From top to bottom, the curves are the inverter flux magnitude, power angle, and inverter output line currents, respectively. The referenced The system steady-state performance is shown in Figure 6. From top to bottom, the curves are values of the inverter flux magnitude and power angle are set to 11 Wb and 0.4 rads, respectively. the inverter flux magnitude, power angle, and inverter output line currents, respectively. The It can bereferenced seen that values by controlling of the inverter the VSIflux withmagnitude SDFC, and both power the inverter angle are flux set amplitudeto 11 Wb and and 0.4 power rads, angle presentrespectively. a large amount It can of be oscillations seen that by and controlling ripples, the though VSI with they SDFC, can trackboth the their inverter references flux amplitude roughly. Hence the lineand currents, power i.e.,angle the present currents a large flowing amount from of theoscillations DG to the and grid, ripples, are seriouslythough they distorted. can track As their explained previously,references these roughly. are the Hence main the disadvantages line currents, i.e., of thethe currents conventional flowing inverter from the flux DG to control the grid, because are the voltageseriously vector applieddistorted. in As every explained sampling previously, period thes is note are necessarily the main disadvantages the most effective of the conventional one. For grid-tied DG systems,inverter the flux injected control currentsbecause the should voltage be vector as sinusoidal applied in as every possible. sampling Otherwise, period is the not power necessarily quality will the most effective one. For grid-tied DG systems, the injected currents should be as sinusoidal as be deteriorated by such distorted current components. Consequently, the grid stability will be affected, possible. Otherwise, the power quality will be deteriorated by such distorted current components. and additionalConsequently, power the lossesgrid stability will be will produced. be affected, On and the additional other hand, power after losses using will the be produced. proposed On PDFC, it can bethe observed other hand, that after the fluxusing amplitude the proposed and PDFC, power it ca anglen be observed are much that better the flux controlled amplitude with and power a significant ripple reduction.angle are much In addition,better controlled the line with currents a significant are also ripple more reduction. sinusoidal In addition, than those the line of currents SDFC. are Toalso further more demonstratesinusoidal than the those better of SDFC. performance of PDFC, Figure7 shows the locus of the inverter To further demonstrate the better performance of PDFC, Figure 7 shows the locus of the inverter flux vector ψV. The horizontal axis represents the α component of the inverter flux vector while the flux vector ΨV. The horizontal axis represents the α component of the inverter flux vector while the vertical axis represents the β component of the inverter flux vector. It is clearly seen that, compared vertical axis represents the β component of the inverter flux vector. It is clearly seen that, compared with SDFC,with SDFC, the tip the of tip the of inverterthe inverter flux flux vector vector ofof PDFCPDFC is is closer closer to toa circle. a circle. While While the aim the of aim inverter of inverter ψ flux controlflux control is to enableis to enable the the inverter inverter flux flux ΨVVto to havehave a a specified specified magnitude magnitude and anda specified a specified position position relativerelative to the to grid the fluxgrid fluxψE, Ψ PDFCE, PDFC shows shows better better abilityability in in regulating regulating |Ψ |Vψ| andV| andδp thanδp SDFC.than SDFC. To obtainTo obtain a better a comparisonbetter comparison in terms in terms of line of currents, line currents, the zoom-in the zoom-in waveforms waveforms and and the harmonicthe spectrumharmonic are plotted spectrum out are in plotted Figure out8. in It canFigure be 8. seen It can that be seen the that current the current when when using using the PDFCthe PDFC strategy is muchstrategy more is sinusoidal much more than sinusoidal that when than usingthat when the using conventional the conventional SDFC method.SDFC method. The totalThe total harmonic harmonic distortion (THD) of the line current for PDFC is 4.09%, much less than the 12.15% for SDFC. distortion (THD) of the line current for PDFC is 4.09%, much less than the 12.15% for SDFC. This is This is a significant improvement of the grid-connected DG system by using the inverter flux control a significantapproach. improvement of the grid-connected DG system by using the inverter flux control approach.
12
11 Flux (Wb) Flux 10 0 0.02 0.04 0.06 0.08 0.1 0.5
0.4 Angle (rads) Angle 0.3 0 0.02 0.04 0.06 0.08 0.1 400 200 0 -200 Current (A) Current -400 0 0.02 0.04 0.06 0.08 0.1 Time (sec) (a)
Figure 6. Cont. Energies 2017, 10, 428 8 of 11 EnergiesEnergies 2017 2017, 10,, 42810, 428 8 of8 11 of 11 Energies 2017, 10, 428 8 of 11 12 12 12
11 11 11 Flux (Wb) Flux Flux (Wb) Flux
Flux (Wb) Flux 10 10 0 0 0.020.02 0.040.04 0.06 0.080.08 0.10.1 10 0 0.02 0.04 0.06 0.08 0.1 0.50.5 0.5 0.4 0.4 0.4 Angle (rads) Angle
Angle (rads) Angle 0.3 0.3 0 0.02 0.04 0.06 0.08 0.1 Angle (rads) Angle 0 0.02 0.04 0.06 0.08 0.1 0.3 4000 0.02 0.04 0.06 0.08 0.1 400 200 200400 0 200 -2000 Current (A) -2000
Current (A) -400 -200 0 0.02 0.04 0.06 0.08 0.1
Current (A) -400 0 0.02 0.04 Time (sec)0.06 0.08 0.1 -400 0 0.02 0.04Time(b) (sec) 0.06 0.08 0.1 Time(b) (sec) Figure 6. Steady-state response. (a) Switching table–based direct flux control (SDFC); (b) Predictive Figure 6. Steady-state response. (a) Switching table–based(b) direct flux control (SDFC); (b) Predictive Figuredirect 6. fluxSteady-state control (PDFC). response. (a) Switching table–based direct flux control (SDFC); (b) Predictive direct flux control (PDFC). directFigure flux 6. Steady-state control (PDFC). response. (a) Switching table–based direct flux control (SDFC); (b) Predictive direct flux15 control (PDFC). 15 15 15 15 10 1015 10 5 105 10 10 (W b) 5 (Wb) 0 05 Vq 5Vq 5 (W b) (Wb) Flux 0Flux -5 -50 Vq Vq (W b) (Wb) 0 0 Vq Vq Flux Flux -5 -10 -10-5 Flux Flux -5 -5 -15 -15 -10 -15 -10 -5 0 5 10 15 -10-15 -10 -5 0 5 10 15 -10 Flux (Wb) -10 Flux (Wb) Vd Vd -15 -15 -15 -10 -5 (a0) 5 10 15 -15 -10 -5 (b) 0 5 10 15 -15 Flux (Wb) -15 Flux (Wb) -15 -10 -5 Vd0 5 10 15 -15 -10 -5 Vd0 5 10 15 Figure 7. The locusFlux of inverter (Wb) flux by using different control methods. Flux(a) SDFC; (Wb) (b) PDFC. (a) Vd (b)Vd 400 (a) (b) Figure 7. The locus of inverter flux by using different control methods. (a) SDFC; (b) PDFC. Figure 7. The locus200 of inverter flux by using different control methods. (a) SDFC; (b) PDFC. Figure 7. The) locus of inverter flux by using different control methods. (a) SDFC; (b) PDFC. 400 0 400
200Current (A ) -200 200 ) 0 -400 0 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 Current (A -200 time (s)
Current (A -200 -400 5 0.06 0.065Fundamental0.07 0.075(50 Hz)= 0.08268.5940.085 THD=12.15%0.09 0.095 0.1 -400 4 0.06 0.065 0.07 0.075 time0.08 (s) 0.085 0.09 0.095 0.1 3 time (s) 5 2 Mag (%) Fundamental (50 Hz)= 268.594 THD=12.15% 5 4 1 Fundamental (50 Hz)= 268.594 THD=12.15% 4 3 0 0 1000 2000 3000 4000 5000 3 2 Mag (%) Frequency (Hz) 2 Mag (%) 1 (a) 1 0 0 1000 2000 3000 4000 5000 0 0 1000 2000Frequency (Hz)3000 4000 5000 Frequency (Hz) (a) (a)
Figure 8. Cont.
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400400
200200
0 0 Current (A) Current (A) -200-200
-400-400 0.060.06 0.0650.065 0.070.07 0.0750.075 0.08 0.0850.085 0.090.09 0.0950.095 0.10.1 timetime (s)
1 1 FundamentalFundamental (50 (50 Hz)=Hz)= 266.895 THD=4.09% THD=4.09% 0.8 0.8 0.6 0.6
Mag (%) 0.4
Mag (%) 0.4 0.2 0.2 0 0 0 1000 2000 3000 4000 5000 0 1000 2000Frequency (Hz)3000 4000 5000 Frequency (Hz) (b) (b) Figure 8. FFT analysis of line currents. (a) SDFC; (b) PDFC. Figure 8. FFT analysis of line currents. ( a) SDFC; ( b) PDFC. 5.2. Dynamic Performance 5.2. Dynamic Performance To further verify the reliability of the proposed inverter controller, the dynamic response is testedTo furtherfurther under verifytheverify command the the reliability reliability of the stepped of of the the proposed change proposed of inverter the inverter flux controller, magnitude controller, the and dynamic the power dynamic responseangle. response The is flux tested is testedunderamplitude theunder command referencethe command of steps the stepped downof the from stepped change 11 Wb ofchange theto 8 flux Wb of magnitude atthe a 0.2flux s, magnitudewhile and the power power and angle. powerangle The reference angle. flux amplitude The steps flux amplitudereferenceup from steps reference0.4 rads down to steps 1.9 from rads down 11 at Wb 0.1 from s to and 811 Wb stepsWb at to adown 8 0.2 Wb s, to whileat − 0.5a 0.2 rads. the s, while powerIt can the be angle seenpower referencefrom angle Figure reference steps 9 that up thesteps from up0.4flux from rads magnitude to0.4 1.9 rads rads to and 1.9 at power 0.1rads s and atangle 0.1 steps scan and reach down steps the todown new−0.5 stateto rads. −0.5 in a Itrads. safe can mannerIt be can seen be without fromseen from Figure causing Figure9 that dangerous 9 thethat flux the fluxmagnitudeovershoot magnitude and currents and power power for angle both angle canmethods. can reach reach However, the the new new stateth statee flux in in aamplitude a safe safe manner manner and without powerwithout angle causingcausing in PDFC dangerous can overshoottrack the currentscurrents reference forfor more both both methods.quickly, methods. especially However, However, during the th fluxe theflux amplitude reference amplitude andstepping and power power down. angle angle Besides, in PDFC in PDFC smaller can track can tracktheactive reference the and reference reactive more quickly, more power quickly, especiallyripples ofespecially PDFC during can theduring also reference be theobserved. reference stepping It is down.worthstepping mentioning Besides, down. smaller Besides, that the active activesmaller and activereactivepower and powerbecomes reactive ripples negative power of ripples PDFC when cantheof PDFC power also be can angle observed. also (δ bep = observed.δV It − is δE worth) is Itcontrolled is mentioning worth tomentioning be thatless thethan that active zero. the This poweractive indicates that the VSI system supplies the power from energy resources to the grid if the inverter flux powerbecomes becomes negative negative when the when power the angle power (δp angle= δV −(δδp E=) δ isV controlled− δE) is controlled to be less to than be less zero. than This zero. indicates This indicatesthatvector the VSI lagsthat system behindthe VSI suppliesthe system grid fluxsupplies the vector. power the from power energy from resourcesenergy resources to the grid to the if thegrid inverter if the inverter flux vector flux vectorlags behind lags behind the grid the flux grid12 vector. flux vector. 10 12 8
10 (Wb) Flux 6 8
Flux (Wb) Flux 6 2 1 2 0 -1
Angle (rads) 1Angle 6 0 x 10 -1
Angle (rads) Angle 2 Q 0 6 x 10 -2 P 2 P,Q (W, Var) (W, P,Q Q 0 1000-2 P
P,Q (W, Var) (W, P,Q 0 1000
Current (A) Current -1000 0.05 0.1 0.15 0.2 0.25 0 Time (sec) (a)
Current (A) Current -1000 0.05 0.1 0.15 0.2 0.25 Time (sec) (a)
Figure 9. Cont.
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12 10 8
Flux (Wb) Flux 6
2 1 0 -1 Angle (rads) 6 x 10 2 Q 0 -2 P P,Q (W, Var) (W, P,Q
1000
0
Current (A) -1000 0.05 0.1 0.15 0.2 0.25 Time (sec) (b)
Figure 9. Dynamic response. (a) SDFC; (b) PDFC. Figure 9. Dynamic response. (a) SDFC; (b) PDFC.
6. Conclusions 6. Conclusions This paper has proposed the PDFC method for VSIs to reduce flux ripple as well as to improve This paper has proposed the PDFC method for VSIs to reduce flux ripple as well as to improve the the current quality. In the proposed PDFC strategy, the system model, together with all the possible current quality. In the proposed PDFC strategy, the system model, together with all the possible inverter inverter switching states, is used to predict the future flux behaviors. The switch state (or voltage switching states, is used to predict the future flux behaviors. The switch state (or voltage vector) that vector) that can minimize the error between the references and the flux amplitude and power angle can minimize the error between the references and the flux amplitude and power angle is then selected is then selected to control the inverter. On the basis of this operation scheme, the inverter flux can be to control the inverter. On the basis of this operation scheme, the inverter flux can be controlled to controlled to have a specified magnitude and a specified position relative to the grid flux. The have a specified magnitude and a specified position relative to the grid flux. The proposed PDFC proposed PDFC method can reduce the inverter flux ripples as well as control the active and reactive method can reduce the inverter flux ripples as well as control the active and reactive powers with a fast powers with a fast transient response and less line current distortions compared with the transient response and less line current distortions compared with the conventional SDFC method. conventional SDFC method. Acknowledgments: Acknowledgments: This work work is is supported supported in part in bypart The by Hong The KongHong Polytechnic Kong Polytechnic University University under Grant under 1-ZE7J. ConflictsGrant 1-ZE7J. of Interest: The author declares no conflict of interest.
ReferencesConflicts of Interest: The author declare no conflict of interest.
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