Analysis and Design of a Double Fuzzy PI Controller of a Voltage Outer Loop in a Reversible Three-Phase PWM Converter

energies

Article Analysis and Design of a Double Fuzzy PI Controller of a Voltage Outer Loop in a Reversible Three-Phase PWM Converter

Weixuan Zhang , Yu Fang * , Rong Ye and Zhengqun Wang

School of Information Engineering, Yangzhou University, Yangzhou 225100, China; [email protected] (W.Z.); [email protected] (R.Y.); [email protected] (Z.W.) * Correspondence: [email protected]; Tel.: +86-158-6132-8650

Received: 15 June 2020; Accepted: 21 July 2020; Published: 23 July 2020

Abstract: Aiming at the application of a reversible three-phase pulse width modulation (PWM) converter with a wide range of AC side voltage and DC side voltage, a double fuzzy proportional integral (PI) controller for voltage outer loop was proposed. The structures of the proposed controller were motivated by the problems that either the traditional PI controller or single fuzzy PI controller cannot achieve high performance in a wide range of AC and DC voltage conditions. The presented double fuzzy controller studied in this paper is a sub fuzzy controller in addition to the traditional fuzzy PI controller; in particular, the sub fuzzy controller can get the auxiliary correction of PI control parameters according to the AC side voltage and the DC side given voltage variation of PWM converter after the reasoning of the sub fuzzy controller, while the traditional fuzzy PI controller outputs the correction of PI control parameters according to the DC voltage error and its error change rate. In this paper, the traditional fuzzy PI controller can be called the main fuzzy controller, and the adaptive adjustment of PI control parameters of the voltage outer loop is the sum of the PI parameter correction output by the main fuzzy controller and the auxiliary PI parameter correction output by the sub fuzzy controller. Finally, the experimental results show that the reversible three-phase PWM converter can achieve excellent dynamic and static performance in a wide range of AC voltage and DC voltage applications by using the proposed double fuzzy PI controller.

Keywords: double fuzzy PI controller; reversible three-phase PWM converter; auxiliary correction of the PI parameters

1. Introduction With the rapid development of power electronic technology in recent years, the reversible three-phase PWM converter circuit has gradually replaced the traditional uncontrolled rectifier circuit such as a diode or thyristor because of its advantages of high power factor, low harmonic content, and energy bidirectional flow [1–3]. Therefore, the three-phase PWM converter has been widely used in a switching mode power supply and especially electric drive fields [4–6]. At present, the traditional double closed-loop PI controller is usually used for a three-phase PWM converter, and the PI parameters are usually obtained for a specific operating condition [7,8]. Considering that one operating condition corresponds to a model, the PI control parameters deduced by the mathematical model are usually the optimal parameters under the corresponding operating condition [9–11]. However, the traditional PI controller cannot adaptively change the control parameters to adapt to the changing operating conditions; therefore, in view of the change of the AC side voltage, the feedforward technology of the AC voltage is presented in reference [12], so that the converter has a certain adaptive ability to the fluctuation of the AC voltage. However, too much feed-forward will

Energies 2020, 13, 3778; doi:10.3390/en13153778 www.mdpi.com/journal/energies Energies 2020, 13, x FOR PEER REVIEW 2 of 19 converterEnergies 2020 has, 13, a 3778 certain adaptive ability to the fluctuation of the AC voltage. However, too much feed2 of 19- forward will lead to system instability, so it is not suitable for a large AC voltage range [13]. For the above reason, many scholars have introduced the fuzzy control technology into the three-phase PWM lead to system instability, so it is not suitable for a large AC voltage range [13]. For the above reason, converter. many scholars have introduced the fuzzy control technology into the three-phase PWM converter. In references [14–16], a fuzzy PI controller for the voltage outer loop of PWM converter is In references [14–16], a fuzzy PI controller for the voltage outer loop of PWM converter is presented. presented. The adaptive correction output by the fuzzy PI controller can be achieved in light with The adaptive correction output by the fuzzy PI controller can be achieved in light with situations situations described fuzzily by the DC bus voltage error and its error change rate, so the PWM described fuzzily by the DC bus voltage error and its error change rate, so the PWM converter can converter can obtain better dynamic and static characteristics under a different load and a sudden obtain better dynamic and static characteristics under a different load and a sudden change of the change of the load compared with PI controller [17,18]. However, the traditional single fuzzy PI load compared with PI controller [17,18]. However, the traditional single fuzzy PI controller does controller does not introduce the information of AC voltage, and as a consequence, the change of AC not introduce the information of AC voltage, and as a consequence, the change of AC voltage cannot voltage cannot adjust the PI parameters to adapt to the new operating situation, especially in the wide adjust the PI parameters to adapt to the new operating situation, especially in the wide range of AC range of AC voltage variation, and the system may not work stably. In reference [19], two types of voltage variation, and the system may not work stably. In reference [19], two types of fuzzy logic fuzzy logic controllers were proposed—that is, in addition to using traditional fuzzy PI controller to controllers were proposed—that is, in addition to using traditional fuzzy PI controller to adjust the adjust the error of controlled variables and its error change rate, the influence of other related error of controlled variables and its error change rate, the influence of other related variables is adjusted variables is adjusted adaptively by the added sub-fuzzy controller. However, in the existing adaptively by the added sub-fuzzy controller. However, in the existing literature, the large number literature, the large number of sub fuzzy controllers are either used to change the quantization factor of sub fuzzy controllers are either used to change the quantization factor or the scale factor—that is, or the scale factor—that is, to realize the self- adjustment of main fuzzy controller parameters [20,21]. to realize the self- adjustment of main fuzzy controller parameters [20,21]. In addition, for the front-end reversible three-phase PWM converter of motor driver power In addition, for the front-end reversible three-phase PWM converter of motor driver power supply, supply, in order to achieve its universality, both the AC side and the DC side have a wide range; that in order to achieve its universality, both the AC side and the DC side have a wide range; that is, is, under the wide range of AC side voltage and DC side voltage application, the reversible three- under the wide range of AC side voltage and DC side voltage application, the reversible three-phase phase PWM converter should work stably and have high dynamic response performance. Obviously, PWM converter should work stably and have high dynamic response performance. Obviously, neither neither the PI controller nor the traditional single fuzzy PI controller is suitable for this application. the PI controller nor the traditional single fuzzy PI controller is suitable for this application. For this For this reason, a double fuzzy PI controller for voltage outer loop is proposed by adding the sub reason, a double fuzzy PI controller for voltage outer loop is proposed by adding the sub fuzzy fuzzy controller to the traditional fuzzy PI controller, and the sub fuzzy controller here generates the controller to the traditional fuzzy PI controller, and the sub fuzzy controller here generates the auxiliary auxiliary correction of PI parameters according to the AC side voltage variation and DC side given correction of PI parameters according to the AC side voltage variation and DC side given voltage voltage variation of the reversible three-phase PWM converter. variation of the reversible three-phase PWM converter. The content of this paper is arranged as follows: first, the double closed-loop control strategy for The content of this paper is arranged as follows: first, the double closed-loop control strategy front-end reversible three-phase PWM converter is analyzed, and the traditional PI controller is for front-end reversible three-phase PWM converter is analyzed, and the traditional PI controller introduced, and then the double fuzzy PI controller is proposed for the application of front-end is introduced, and then the double fuzzy PI controller is proposed for the application of front-end reversible three-phase PWM converter with wide range of AC side voltage and DC side voltage in reversible three-phase PWM converter with wide range of AC side voltage and DC side voltage motor driver power supply, and the design methodology of double fuzzy PI controller is given. in motor driver power supply, and the design methodology of double fuzzy PI controller is given. Finally, the experimental results verify the feasibility and correctness of the proposed double fuzzy Finally, the experimental results verify the feasibility and correctness of the proposed double fuzzy PI controller. PI controller.

2.2. Circuit Circuit and and Control Control Strategy Strategy of of Reversible Reversible Three Three-Phase-Phase PWM PWM Converter Converter TheThe current motor driver power supply is generally composed of two back back-to-back-to-back reversible threthree-phasee-phase PWM PWM converters converters as as shown shown in in Figure Figure 11.. The ACAC sideside ofof thethe front-endfront-end PWMPWM converterconverter isis connectedconnected to to the the grid grid or or AC AC microgrid, microgrid, and and its its DC DC side side provides provides bus voltage for the back back-end-end PWM converter.converter. The The front front-end-end PWM PWM converter converter can be be used used not not only only as as a a rec rectiﬁertifier to to provide provide DC DC voltage voltage for for thethe back back-end-end PWM PWM converter converter but but also also as as an an inverter inverter to to realize realize energy energy feedback feedback toward toward the the AC AC side side inin the the process process of of motor braking and and deceleration [2 [222].]. In In this this paper, paper, the the controller controller of of the the front front-end-end reversiblereversible three three-phase-phase P PWMWM converter in Figure 1 1 is is studied.studied.

Front End Back End

DC Side M

AC PWM PWM AC Side Converter Converter Motor

Motor driver power supply Figure 1. System structure diagram of motor driver power supply.

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Energies 2020, 13, 3778 3 of 19 Figure 1. System structure diagram of motor driver power supply.

InIn thisthis paper,paper, thethe circuitcircuit topologytopology ofof reversiblereversible three-phasethree-phase PWMPWM converterconverter atat thethe frontfront endend ofof motormotor driver driver power power supply supply adopts adopts neutral neutral point point clamped clamped (NPC) (NPC) I-type I three-phase-type three- three-levelphase three circuit,-level ascircuit, shown as inshown Figure in2 ,Figure which 2, mainly which includes mainly includes the following the following three parts: three AC parts: side, AC bridge side, circuit, bridge and circuit DC , and DC side. usa, usb, and usc are the phase voltages of the AC side with the midpoint of N; Ls and Rs side. usa, usb, and usc are the phase voltages of the AC side with the midpoint of N; Ls and Rs are the are the inductance and the equivalent resistance of the AC side, respectively; isa, isb, and isc are the inductance and the equivalent resistance of the AC side, respectively; isa, isb, and isc are the currents atcurrents the AC at side. the AC Each side phase. Each of phase the three-phase of the three bridge-phase arm bridge is composed arm is composed of four insulated of four insulated gate bipolar gate bipolar transistor (IGBT) switches Si1~Si4 (i = a,b,c) with freewheel diodes and two clamping diodes transistor (IGBT) switches Si1~Si4 (i = a,b,c) with freewheel diodes and two clamping diodes Di1~Di2 Di1~Di2 (i = a,b,c). Two bus capacitors (Cd1 and Cd2) of the same size are connected in series at the DC (i = a,b,c). Two bus capacitors (Cd1 and Cd2) of the same size are connected in series at the DC side. side. O is the midpoint of the capacitance, and Udc is the DC bus voltage. O is the midpoint of the capacitance, and Udc is the DC bus voltage.

Sa1 Sb1 Sc1

Cd1 Sa2 Sb2 Sc2 - s usa Ls R isa Da1 Db1 Dc1 ~ A +

usb Ls Rs isb Udc N ~ B O - usc Ls Rs isc ~ C

Sa3 Sb3 Sc3 + AC Side Da2 Db2 Dc2 Cd2 - a4 b4 S S Sc4

Bridge circuit DC Side

FigureFigure 2. 2. Main circuit of neutralneutral pointpoint clampedclamped (NPC)(NPC) I-typeI-type three-phase three-phase three-level three-level pulsepulse width width modulationmodulation (PWM)(PWM) converter.converter.

TheThe PWM PWM converter converter adopts adopts the the double double closed-loop closed-loop control control structure structure based based on the onAC the side AC voltage side vectorvoltage oriented vector oriented control, including control, including the voltage the outer voltage loop outer and the loop current and the inner current loop, inner as shown loop, in as Figure shown3. Thein Figure modulation 3. The methodmodulation adopts method space adopts vector s pulsepace v widthector p modulationulse width m (SVPWM).odulation The(SVPWM orientation). The angleorientationθ calculated angle θ by calculated phase-locked by phase loop- (PLL)locked is loop sent (PLL) to abc is/dq sent transformation to abc/dq transformation unit and inverse unit Park and transformationinverse Park transformation (I-Park) unit. Through(I-Park) unit. abc/dq Through transformation, abc/dq transformation, the mathematical the model mathematical of the system model is transformedof the system from is transformed three-phase from static three ABC-phase coordinate static system ABC coordinate to dq rotation system coordinate to dq rotation system. coordinateud and uq aresystem. the voltage ud and componentsuq are the voltage of the componentsd-axis and q-axis, of the respectively, d-axis and andq-axisid ,and respectively,iq are current and components id and iq are ofcurrend-axist components and q-axis, respectively. of d-axis and The q- currentaxis, respectively. components T ofhe thecurrentd-axis components and q-axis are of mutually the d-axis coupled and q- afteraxis are coordinate mutually transformation. coupled after coordinate For this reason, transformation. the current feed-forwardFor this reason, decoupling the current control feed- strategyforward isdecoupling introduced, control in which strategy the components is introduced,ωLs iinq and whichωL stheid of componentsq-axis and d -axisωLsiq areand injected ωLsid of into q-axis the andd-axis d- andaxis qare-axis injected forward into channels the d-axis of and the currentq-axis forward loop, respectively, channels of so the that current the independent loop, respectively, control so of that the currentthe indepe componentsndent control of the ofdq theaxis current can be components realized [23 of]. the dq axis can be realized [23].

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usa isa Ls Rs Cd1 + usb isb Ls Rs N Udc usc isc Ls Rs Cd2 -

S S S ud a b c abc abc drive circuit d-q d-q q u iq id SVPWM θ PLL I-Park _ *  _ _ iq  0 PI ωLsid  Traditional PI uq controller _ * i _  *  Single fuzzy d  PI Udc _ ωLsiq PI controller  Udc ud The proposed double fuzzy PI controller Controller for voltage Outer Loop

Figure 3. DoubleDouble closed closed-loop-loop control structure of three three-phase-phase PWM converter.

The current inner loop is usually designed as a follower with PI controller to follow the current The current inner loop is usually designed as a follower with PI controller to follow the current given by the voltage outer loop, and track the voltage phase of the AC voltage through voltage given by the voltage outer loop, and track the voltage phase of the AC voltage through voltage vector- vector-oriented control, so as to realize the unity power factor and sinusoidal current at the AC oriented control, so as to realize the unity power factor and sinusoidal current at the AC side [24,25]. side [24,25]. In order to realize the stable DC side voltage (Udc) control without static error, it is In order to realize the stable DC side voltage (Udc) control without static error, it is necessary to design necessary to design the controller for voltage outer loop, as shown in Figure3, the voltage outer loop the controller for voltage outer loop, as shown in Figure 3, the voltage outer loop compensator can compensator can be implemented by the traditional PI controller, traditional single fuzzy PI controller, be implemented by the traditional PI controller, traditional single fuzzy PI controller, and the and the proposed double fuzzy PI controller in this paper. Different from the current inner loop, proposed double fuzzy PI controller in this paper. Different from the current inner loop, the response the response speed of the voltage outer loop is much slower, and the cut-off frequency of the voltage speed of the voltage outer loop is much slower, and the cut-off frequency of the voltage loop should loop should be far less than that of the current loop [26]. Therefore, when the traditional PI controller be far less than that of the current loop [26]. Therefore, when the traditional PI controller is used for is used for system correction, the anti-disturbance performance of the voltage outer loop should be system correction, the anti-disturbance performance of the voltage outer loop should be mainly mainly considered [27]. considered [27]. The parameters of PWM converter in Figure3 are as follows: Rs = 0.002 Ω, Ls = 1 mH, The parameters of PWM converter in Figure 3 are as follows: Rs = 0.002 Ω, Ls = 1 mH, Cd1 = Cd2 = Cd1 = Cd2 = 1475 µF. AC side phase voltage is E = 220 V, the DC side voltage is Udc = 700 V, the power 1475 μF. AC side phase voltage is E = 220 V, the DC side voltage is Udc = 700 V, the power is Po = 16 is Po = 16 kW, and the switching frequency is f s = 20 kHz. Bode diagram method is used to design kW, and the switching frequency is fs = 20 kHz. Bode diagram method is used to design PI parameters PI parameters of the voltage outer loop with a cut-off frequency of 100 Hz and a phase margin of of the voltage outer loop with a cut-off frequency of 100 Hz and a phase margin of 45°. Considering 45◦. Considering the sampling period with digital control, the calculated PI control parameters are the sampling period with digital control, the calculated PI control parameters are kp = 5.587, ki = k k 0.05892.p = 5.587, It isi = noticed0.05892. that It is noticedthe parameters that the parameters of traditional of traditional PI controller PI controller are the optimalare the optimal values valuestheoretically theoretically under under some some certain certain working working conditions, conditions, which which are are largely largely dependent dependent on on the mathematical model model of of some some certain certain operating operating point. point. However, However, with with the thetraditional traditional PI controller, PI controller, the thedesigned designed PI control PI control parameters parameters cannot cannot be modified be modified at will at will [28] [28, even], even if the if the traditional traditional single single fuzzy fuzzy PI PIcontroller controller is used is used in inthe the Figure Figure 3,3 ,the the PI PI parameters parameters can can only only be be adaptively adaptively changed according toto thethe DC voltage error andand its error changechange rate.rate. So it is didifficultfficult for the front-endfront-end reversiblereversible three-phasethree-phase PWM converter with a wide range of AC side voltage and DC side voltage to obtain better control performance if the traditional PI controller and the single fuzzy PI controllercontroller is used in the voltage outer loop. Therefore, thethe double fuzzy PI control method for the voltage outer loop is proposed on the basis of thethe traditionaltraditional singlesingle fuzzy fuzzy PI PI controller, controller, which which can can adaptively adaptively adjust adjust PI parametersPI parameters to improveto improve the dynamicthe dynamic and static and stperformanceatic performance of the front-end of the front reversible-end reversible three-phase three PWM-phase converter PWM according converter toaccording a variety to of a circumstances.variety of circumstances. The following The following will discuss will the discuss design the method design of method the double of the fuzzy double PI controllerfuzzy PI controller for the voltage for the outer voltage loop. outer loop.

3. Analysis and Design Method of the Double Fuzzy PI Controller for Voltage Outer Loop

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3. AnalysisAs shown and in Design Figure Method4, the proposed of the Double doubleFuzzy fuzzy PIPI Controllercontroller is for implemented Voltage Outer by adding Loop a sub fuzzyAs controller shown in to Figure the 4 traditional, the proposed fuzzy double PI controller, fuzzy PI controller and the sub is implemented fuzzy controller by adding outputs a sub the fuzzyauxiliary controller correction to the of traditionalPI parameters. fuzzy Considering PI controller, the and application the sub fuzzy of a controllerwide range outputs of AC thevoltage auxiliary and correctionDC voltage of in PI a parameters. reversible three Considering-phase PWM the application converter, of the a wide AC rangeside voltage of AC voltage variation and and DC the voltage DC inside a reversiblegiven voltage three-phase variation PWM are taken converter, as the the inputs AC side of the voltage sub fuzzy variation controller, and the and DC sidethe outputs given voltage of the variationsub fuzzy are controller taken as is the designed inputs of as the the sub auxiliary fuzzy controller,correction and of the the PI outputs parameters of the of sub the fuzzy voltage controller outer isloop; designed the traditional as the auxiliary fuzzy PI correction controller ofpresented the PI parameters here is called of thethe voltagemain fuzzy outer controller loop; the and traditional outputs fuzzythe correction PI controller of PI presentedcontrol parameters here is called according the main to the fuzzy DC controller voltage error and and outputs its error the correctionchange rate. of PIAnd control the adaptive parameters adjustment according of toPI thecontrol DC voltageparameters error of and the itsvoltage error changeouter loop rate. is Andthe sum the adaptiveof the PI adjustmentparameter correction of PI control output parameters by the ofmain the voltagefuzzy controller outer loop and is thethe sumauxiliary of the PI PI parameter parameter correction correction output by thethe mainsub fuzzy fuzzy controller. controller The and output the auxiliary of the PIPI parameterregulator in correction the Figure output 4 serves by the as the sub d fuzzy-axis * * controller.current given The signal output id of of the current PI regulator inner loop, in the and Figure G(s)4 is serves the transfer as the d-axisfunction current from giventhe d- signalaxis currentid of currentto the DC inner voltage loop, U anddc. G(s) is the transfer function from the d-axis current to the DC voltage Udc.

Main fuzzy controller Fuzzy Fuzzification rule Defuzzification reasoning

Δkp1 Δki1 ec * de/dt id* Udc Udc e PI regulator Gs() + - Udc

* Δkp2 Δki2 Udc0 - Sub fuzzy controller + ΔU * dc Fuzzy Fuzzification Defuzzification uac_rms Δuac_rms rule + reasoning -

uac_rms0 The Proposed Double Fuzzy PI Controller Figure 4. TheThe block block diagram of voltage outer loop with the proposed d doubleouble fuzzy PI controller.controller.

* The inputs of the mainmain fuzzyfuzzy controllercontroller areare thethe errorerror ee betweenbetween thethe DCDC busbus givengiven voltagevoltageU Udcdc* and the actual DCDC sideside voltagevoltage UUdcdc,, and the errorerror changechange raterate ecec.. After fuzzification,fuzzification, reasoning by fuzzy rules, rules, and and then then after after defuzzification, defuzzification, a agroup group of ofcorrection correction values values of PI of control PI control parameters parameters for forPI regulator PI regulator,, Δkp1∆ andkp1 andΔki1 are∆ki1 obtained;are obtained; the inputs the inputs of the of sub the fuzzy sub fuzzycontroller controller are the are change the change of the * ofeffective the eff ectivevalue of value AC ofphase AC voltage phase voltage Δuac_rms∆ anduac_rms the andchange the of change the DC of gi theven DC voltage given Δ voltageUdc*. Through∆Udc . Throughthe same theworking same workingprocedure procedure as the main as the fuzzy main controller fuzzy controller,, another another group groupof correction of correction values values of PI ofcontrol PI control parameters parameters,, Δkp2 ∆andkp2 and Δki2 ∆areki2 are obtained obtained and and here here defined defined as as auxiliary auxiliary correction correction of of PI parameters for for PI regulator. PI regulator Adding. Adding the above thetwo above groups two of PI groups parameter of corrections PI parameter correspondingly corrections tocorrespondingly get the sums, i.e., to ∆ getkp1 the+ ∆ sums,kp2 and i.e.,∆k i1 Δk+p1∆ +k i2Δ,k thenp2 and adding Δki1 + these Δki2, sums then to adding the previous these sums values to of the PI parameters,previous values the currentof PI parameters values of, PI the parameters current valueskp and of kPIi are parameters calculated. kp and ki are calculated.

3.1. Design of Main Fuzzy Controller The structure of the main fuzzy controller is shown in FigureFigure4 4.. TheThe mainmain designdesign processprocess includesincludes three parts—fuzzification,parts—fuzzification, fuzzyfuzzy rulerule reasoningreasoning andand defuzzification.defuzzification. 3.1.1. Fuzzification 3.1.1. Fuzzification Fuzzification is to convert precise input and output values into fuzzy linguistic values. The concrete Fuzzification is to convert precise input and output values into fuzzy linguistic values. The implementation steps are as follows: concrete implementation steps are as follows: * 1) The basic universe of error e between the DC side given voltage Udc and the sampling Udc, (1) The basic universe of error e between the DC side given voltage Udc* and the sampling Udc, and the basic universe of the corresponding error change rate ec can be obtained by simulation and and the basic universe of the corresponding error change rate ec can be obtained by simulation and experiment. Here, the range of error e is [−600, +600], and the range of error change rate ec [−300, +300]. Assuming that the fuzzy universe of error e and error change rate ec is set as [−3, 3], then the two

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Energies 2020, 13, x FOR PEER REVIEW 6 of 19 experiment. Here, the range of error e is [ 600, +600], and the range of error change rate ec [ 300, +300]. − − Assumingintervals are that discretized, the fuzzy and universe the discrete of error sete andis {−3, error −2, − change1, 0, 1, 2, rate 3}.ec Hence,is set asthe [ error3, 3], e thenand the the error two − intervalschange rate are discretized,ec can be transformed and the discrete from setthe is basic { 3, universe2, 1, 0, to 1, 2,the 3}. fuzzy Hence, universe the error bye theand follo the errorwing − − − changequantization rate ec factorcan be: transformed from the basic universe to the fuzzy universe by the following quantization factor: ( ke 3 / 600  0.005 ke = 3/600 = 0.005 (1) k 3 / 300 0.01 (1) kec =ec 3/300 = 0.01

The outputs of the main fuzzy controller are a group of correction values, Δkp1 and Δki1, The outputs of the main fuzzy controller are a group of correction values, ∆kp1 and ∆ki1, respectively,respectively, correspondingcorresponding toto thethe proportionalproportional controlcontrol parametersparameters andand integralintegral controlcontrol parametersparameters inin thethe PIPI regulator.regulator. TheThe rangesranges ofof thethe twotwo outputsoutputs areare alsoalso obtainedobtained byby simulationsimulation andand experiment,experiment, here the range of Δkp1 is [−1.5, +1.5], while Δki1 [−7.5 × 10−33, +7.5 × 10−3]. Assuming that the fuzzy here the range of ∆kp1 is [ 1.5, +1.5], while ∆ki1 [ 7.5 10− , +7.5 10− ]. Assuming that the fuzzy universe of Δkp1 and Δki1 −is also set as [−3, 3], the− discre× te set is {−3,× −2, −1, 0, 1, 2, 3}. The following universe of ∆kp1 and ∆ki1 is also set as [ 3, 3], the discrete set is { 3, 2, 1, 0, 1, 2, 3}. The following scale factor can be gotten to transform Δ−kp1 and Δki1 from basic universe− − −to fuzzy universe: scale factor can be gotten to transform ∆kp1 and ∆ki1 from basic universe to fuzzy universe:    ( kc1 1.5 / 3 0.5  = = (2) kc1 1.5/333 0.5  kc2 7.5 103 / 3 2.5 103 (2) kc = 7.5 10 /3 = 2.5 10 2 × − × − (2) Considering that the input voltage error e and error change rate ec, as well as the output Δkp1 and 2)Δk Consideringi1 can change that in positive the input and voltage negative error directions,e and error the change fuzzy rate universeec, as well is described as the output by seven∆kp1 andlinguistic∆ki1 can values: change “Negative in positive Big (NB) and, negative” “Negative directions, Middle ( theNM fuzzy),” “Negative universe Small is described (NS),” “Z byero seven (ZE),” linguistic“Positive values:Small (PS) “Negative,” “Positive Big (NB),”Middle “Negative (PM),” and Middle “Positive (NM),” Big“Negative (PB).” Then Small, the (NS),”quantized “Zero discrete (ZE),” “Positiveset {−3, −2, Small −1, 0, (PS),” 1, 2, “Positive3} can be Middledescribed (PM),” as fuzzy and language “Positive set Big {N (PB).”B, NM Then,, NS, the ZE quantized, PS, PM, P discreteB}. The set { 3, 2, 1, 0, 1, 2, 3} can be described as fuzzy language set {NB, NM, NS, ZE, PS, PM, PB}. membership− − − function of input and output is defined as a triangular membership function, as shown Thein Figure membership 5. function of input and output is deﬁned as a triangular membership function, as shown in Figure5.

Membership NB NM NS ZE PS PM PB 1

0.5

-3 -2 -1 0 1 2 3

FigureFigure 5. 5.Triangular Triangular membership membership function function of ofe ,eec,ecand and∆ Δkp1kp1,,Δ∆kki1i1.

ThroughThrough the the aboveabove two two steps, steps, four four precise precise signalssignals of of input input andand output output can can be be fuzziﬁed fuzzified andand transformedtransformed intointo fuzzyfuzzy languagelanguage set.set.

3.1.2.3.1.2. FuzzyFuzzy Rule ReasoningReasoning TheThe designdesign ofof fuzzyfuzzy rulesrules ofof mainmain fuzzyfuzzy controllercontroller willwill directlydirectly aaffectffect thethe parametersparameters ofof thethe PIPI controllercontroller andand then then a ffaffectect the the dynamic dynamic and and static static performance performance of the of PWM the PWM converter. converter. In order In toorder obtain to betterobtaink pbetterand k ki pthrough and ki through the fuzzy the controller, fuzzy controller, it is necessary it is necessary to analyze to trend analyze of PI trend parameters of PI parameters with error ewithand error changee and error rate changeec. First, rate the ec influence. First, the of influence PI parameters of PI on parameters the dynamic on andthe dynamic static performance and static ofperformance the system of is analyzed:the system is analyzed:

1)1) Proportional coefficient coeﬃcient kkpp——whenwhen kpk pis is small, small, the the response response speed speed of the of the system system is slow, is slow, the adjustmentthe adjustment time time is long, is long, the the regulation regulation accuracy accuracy is islow, low, and and the the dynamic dynamic and and static performance of the system poor; with thethe increaseincrease ofof kkpp, thethe systemsystem responseresponse speedspeed isis accelerated,accelerated, andand thethe system regulation accuracy is increased, but when kp is too large, the system will produce large overshoot, and the oscillation times will increase, the adjustment time will become longer, and even leading to the system instability in serious cases.

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system regulation accuracy is increased, but when kp is too large, the system will produce large overshoot, and the oscillation times will increase, the adjustment time will become longer, Energies 2020, 13, x FOR PEER REVIEW 7 of 19 and even leading to the system instability in serious cases. k k 2) InteIntegralgral coecoefficientfficient ik—wheni—when ikisi is small, small, the the output output static static error oferror the system of the issystem difficult is to difficult eliminate, to k eliminate,which leads which to the leads low to adjustment the low adjustment accuracy of accuracy the system; of the with system; the increase with the of increasei, the elimination of ki, the k eliminationspeed of static speed error of static of the error system of the is accelerated,system is accelerated, but if i is bu toot if large, ki is too the large, system the will system produce will produceintegral saturationintegral saturation at the initial at the stage initial of response,stage of response, causing large causing overshoot, large overshoot, and even leadingand even to leadingthe system to the oscillating, system oscillating, which will which affect will the stabilityaffect the of stability the system of the [29 system]. [29].

According to the influence inﬂuence of kp andand kkii onon the the dynamic dynamic and and static static performance performance of of the the system, system, the fuzzyfuzzy rule table of thethe mainmain fuzzyfuzzy controllercontroller isis designed. designed. TheThe step step response response curve curve is is shown shown in in Figure Figure6. 6.The The requirements requirements of PI of parameters PI parameters of the of controlled the controlled object in object diﬀerent in different regions are regions analyzed are according analyzed accordingto the range to ofthe error rangee and of error error e changeand error rate changeec. The rate analysis ec. The is analysis as follows: is as follows:

1) abab sectionsection—if—if ee andand ecec areare positive positive big then the larger kp andand the smaller kii shouldshould be be needed in orderorder to to achieve achieve faster faster response response speed speed and and suppress suppress occurrence occurrence of large overshoot;

2) bcbc sectionsection—if—if e is positive mediummedium andandec ecis is positive positive big, big then, then larger largerkp k,p medium, mediumki karei are selected selected in inorder order to to reach reach stable stable value value as as soon soon as as possible; possible; p 3) cdcd sectionsection—if—if ee changeschanges from from positive positive small small value value to to 0 0 and and ecec is positive big big,, then the larger kp and ki should be selected in order to implement the stability of the system; and ki should be selected in order to implement the stability of the system; 4) dede sectionsection—if—if ee changeschanges from from 0 0 to to negative negative small small value value and and ecec changeschanges from from positive positive big big value to to 0, then larger kp and medium ki are needed in order to suppress the overshoot; 0, then larger kp and medium ki are needed in order to suppress the overshoot;

5) efef sectionsection—if—if ee isis from from negative negative small small (or (or negative negative middle) middle) value value to to 0 0 and and ecec isis from from 0 0 to to negative negative middle value, then the smaller kp should be chosen and ki gradually increases from positive middle value, then the smaller kp should be chosen and ki gradually increases from positive middlemiddle value value to to positive positive big big value value in in order order to to reduce reduce or or prevent prevent secondary secondary oscillation; 6) fg section—if e is from 0 to positive small (or positive middle) value and ec from negative big 6) fg section—if e is from 0 to positive small (or positive middle) value and ec from negative big value to negative small value, then kp should be increased gradually and ki should be decreased value to negative small value, then kp should be increased gradually and ki should be decreased gradually in order to speed up the system response and prevent overshoot; gradually in order to speed up the system response and prevent overshoot; 7) gh section—if e decreases gradually from positive small (or positive middle) value to 0 and ec 7) gh section—if e decreases gradually from positive small (or positive middle) value to 0 and ec decreases slowly, then kp and ki should be increased gradually in order to make the system have decreases slowly, then k and k should be increased gradually in order to make the system have good stability. p i good stability.

y e d h c f g

t Figure 6. StepStep response response curve curve..

According to the above analysis, the fuzzy languagelanguage rulerule tablestables ofof ∆Δkkp1p1 and ∆Δki1i1 areare designed designed as shownshown in Table Tables 11 andand 2Table, respectively. 2, respectively.

Table 1. Fuzzy language rules of Δkp1.

ec NB NM NS ZE PS PM PB e

NB PB PB PB PB PS ZE NS

NM PB PM PM PM ZE NS NM

Energies 2020, 13, 3778 8 of 19

Table 1. Fuzzy language rules of ∆kp1. ec NB NM NS ZE PS PM PB e NB PB PB PB PB PS ZE NS NM PB PM PM PM ZE NS NM NS PM PS PM PM NS ZE NB ZE ZE ZE PS PS PS ZE ZE PS NB NM NS PM PM PS PM PM NM NS ZE PM PM PM PB PB NS ZE PS PB PB PB PB

Table 2. Fuzzy language rules of ∆ki1. ec NB NM NS ZE PS PM PB e NB NB NB NB PB PM PS ZE NM NB NB NM PB PS ZE NS NS NB NM NM PM ZE NS NM ZE ZE ZE PS ZE NS NM NM PS NM NS ZE NS ZE NM NB PM NS ZE PS NM PS NB NB PB ZE PS PM NM PM NB NB

Any rule in the fuzzy language rules table of ∆kp1 and ∆ki1 can be expressed in the form of “Ri: if e is Ai and ec is Bi then u is Ci.” Then, the statement of fuzzy language rules table can be expressed as follows:

1)R 1: if e is NB and ec is NB then ∆kp1 is PB and ∆ki1 is NB 2)R 2: if e is NB and ec is NM then ∆kp1 is PB and ∆ki1 is NB 3)R 3: if e is NB and ec is NS then ∆kp1 is PB and ∆ki1 is NB ...

49) R49: if e is PB and ec is PB then ∆kp1 is PB and ∆ki1 is NB For the membership function with seven linguistic values corresponding to the two inputs, the total number of the control rules is 7 7 = 49. Each fuzzy control rule can give a fuzzy implication × relation Ri (i = 1, 2, ... , 49). Then, the total fuzzy implication relations corresponding to fuzzy rules are as follows: R = R R ... R (3) 1 ∪ 2 ∪ 49 3.1.3. Defuzziﬁcation

The result of fuzzification and fuzzy reasoning is the fuzzy set of ∆kp1 and ∆ki1, so it is necessary to defuzzify the fuzzy set in order to get a clear and precise values of the correction. Here, the weighted average method is used to defuzzify. For each element xi (i = 1, 2, ... , n) in the universe, the membership degree u(i) of the fuzzy set after fuzzification is taken as the weighting coefficient of the corresponding element, and the final decision value xo can be obtained from the following Equation (4):

Pn xi u(i) i=1 · x0 = (4) Pn u(i) i=1 Energies 2020, 13, x FOR PEER REVIEW 9 of 19

n  xi  u() i i1 x0  n (4)

Energies 2020, 13, 3778 ui() 9 of 19 i1

After defuzzification, Δkp1 and Δki1 are transformed from fuzzy universe to basic universe throughAfter the defuzzification, scale factor introduced∆kp1 and ∆ inki1 Sectionare transformed 3.1.1, and fromthe final fuzzy actual universe output to basic value universe is obtained through. the scale factor introduced in Section 3.1.1, and the final actual output value is obtained. 3.2. Design of the Sub Fuzzy Controller 3.2. Design of the Sub Fuzzy Controller The sub fuzzy controller is designed for the changes of AC voltage and DC given voltage, and the PIThe parameters sub fuzzy can controller be further is designed adjusted. for The the changesmain design of AC flow voltage of the and sub DC fuzzy given controller voltage, and is the PIsame parameters as that of can the be main further fuzzy adjusted. controller, The so main it will design not be flow explained of the sub in fuzzydetail. controller is the same as that ofThe the 16 main kW fuzzyreversible controller, three-phase so it will PWM not con be explainedverter in this in detail. paper is developed: the range of the AC phaseThe 16 voltage kW reversible effective three-phase value is 20 PWM0~24 converter0 V (line-to in- thisneutral paper) and is developed: the range theof DC range side of’s the given AC phasevoltage voltage is 650~75 effective0 V. In value the experimental is 200~240 V prototype, (line-to-neutral) the rated and AC the phase range voltage of DC side’s effective given value voltage is 22 is0 650~750V, and the V. Inrated the DC experimental voltage is prototype,700 V. If the the rated rated AC AC phase phase voltage voltage effective effective value value of is 22 2200 V V, is and taken the ratedas the DC midpoint voltage u isac_rms 7000 V.of Ifthe the range rated of AC the phase AC voltagephase voltage effective effective value of value, 220 V isthen taken the as basic the midpointuniverse uofac_rms0 AC phaseof the voltage range effective of the AC value phase variation voltage Δ euffac_rmsective is [ value,−20, 20], then and the if the basic quantization universe of factor AC phaseis 0.1, voltage effective value variation ∆uac_rms is [ 20, 20], and if the quantization factor is 0.1, then the then the fuzzy universe is [−2, 2] and the discrete− set is {−2, −1, 0, 1, 2}. The corresponding five-section fuzzy universe is [ 2, 2] and the discrete set is { 2, 1, 0, 1, 2}. The corresponding five-section fuzzy fuzzy language variables− are {NB, NS, ZE, PS, PB}.− −If the rated DC side voltage value 700 V is taken languageas the midpoint variables Udc are*, then {NB, the NS, basic ZE, universe PS, PB}. of If the ratedDC given DC sidevoltage voltage variation value Δ 700Udc* V is is [− taken50, 50], as and the midpoint U *, then the basic universe of the DC given voltage variation ∆U * is [ 50, 50], and if the if the quantizationdc factor chooses 0.04, then the fuzzy universe is [−2, 2] and dcthe discrete− set is {−2, −1, quantization factor chooses 0.04, then the fuzzy universe is [ 2, 2] and the discrete set is { 2, 1, 0, 1, 2}. 0, 1, 2}. The corresponding five fuzzy language variables− are {NB, NS, ZE, PS, PB}.− −Here the Thememberships corresponding functions five fuzzyof Δuac_rms language and Δ variablesUdc* can choose are {NB, the NS, same ZE, triangular PS, PB}. Here membership the memberships function * functionswhich are of shown∆uac_rms in Figureand ∆U 7.dc can choose the same triangular membership function which are shown in Figure7.

Membership NB NS ZE PS PB 1

0.5

-2 -1 0 1 2

** FigureFigure 7. 7.Triangular Triangular membership membership function function of of∆ Δuuac_rmsac_rms and Δ∆Udc . .

Through thethe designdesign methodmethod of of the the traditional traditional PI PI controller controller for for the the voltage voltage outer outer loop loop introduced introduced in Sectionin Section2, the 2, optimal the optimal PI parameters PI parameters in the Tables in the3 Tableand4 scan 3 and be obtained, 4 can be respectively, obtained, respectively by calculation,, by * simulation,calculation, andsimulation experiment, and experiment under diﬀerent under AC different voltage AC and voltage DC given and voltage DC given conditions. voltage conditionsUdc is the. DCUdc* busis the given DC bus voltage given and voltageUac_rms andis theUac_rms eﬀective is the valueeffective of ACvalue phase of AC voltage. phase voltage.

Table 3.3. TheThe optimaloptimal valuevalue ofof kkpp.

* * Udc U650dc 675 700 725 750 Uac_rms 650 675 700 725 750 Uac_rms 200 5.76 5.986 6.213 6.44 6.666 200 5.76 5.986 6.213 6.44 6.666 210 5.453 5.668 5.883 6.098 6.312 210 5.453 5.668 5.883 6.098 6.312 220 5.179220 5.179 5.3835.383 5.587 5.587 5.791 5.994 5.791 5.994 230 4.931230 4.931 5.1255.125 5.319 5.319 5.514 5.708 5.514 5.708 240 4.706240 4.706 4.8914.891 5.077 5.077 5.262 5.447 5.262 5.447

Energies 2020, 13, 3778 10 of 19

Table 4. The optimal value of ki.

* Udc 650 675 700 725 750 Uac_rms 200 0.05924 0.06143 0.06363 0.06583 0.06803 210 0.05698 0.05909 0.06121 0.06332 0.06544 220 0.05486 0.05689 0.05892 0.06096 0.063 230 0.05285 0.05481 0.05678 0.05874 0.06071 240 0.05097 0.05286 0.05475 0.05665 0.05855

The trend of PI parameters changing with the variation of AC voltage and DC given voltage can be found in Tables3 and4. As a consequence, the fuzzy rules table of output value ∆kp2 and ∆ki2 can be deduced as shown in Tables5 and6. ∆uac_rms is the change of the eﬀective value of AC phase * voltage and ∆Udc is the change of the DC given voltage.

Table 5. Fuzzy language rules of ∆kp2.

* ∆Udc NB NS ZE PS PB ∆Uac_rms NB ZE PS PB PB PB NS ZE ZE PS PS PB ZE NS ZE ZE ZE PS PS NS NS ZE ZE ZE PB NB NB NS ZE ZE

Table 6. Fuzzy language rules of ∆ki2.

* ∆Udc NB NS ZE PS PB ∆Uac_rms NB ZE ZE PS PB PB NS ZE ZE PS PS PB ZE NS ZE ZE PS PB PS NB NS ZE ZE PS PB NB NB NS ZE ZE

If the basic universe of ∆k is defined as [ 1, 1] and the scale factor is 0.5, then the fuzzy universe p2 − is [ 2, 2], the corresponding discrete set is { 2, 1, 0, 1, 2}, and the corresponding five fuzzy language − − − variables are {NB, NS, ZE, PS, PB}. If the basic universe of ∆k is defined as [ 10 3, 10 3] and the scale i2 − − − factor is 0.5 10 3, then the fuzzy universe is [ 2, 2], the corresponding discrete set is { 2, 1, 0, 1, 2}, × − − − − and the corresponding five fuzzy language variables are {NB, NS, ZE, PS, PB}. Here, the triangular membership functions of ∆kp2 and ∆ki2 are the same as that of the input values in Figure7. Reasoning based on the fuzzy rules in Tables5 and6, the output fuzzy language variables ∆kp1 and ∆ki1 can be obtained. As with the main fuzzy controller, the weighted average method is used to defuzzify. After defuzzification the actual auxiliary correction of PI control parameters ∆kp2 and ∆ki2 are obtained through the precision with the scale factor. Finally, the auxiliary correction values of PI control parameters ∆kp2 and ∆ki2 obtained by the sub fuzzy controller are added to the correction values of PI parameter ∆kp1 and ∆ki1 obtained by the main fuzzy controller, respectively, and we can get the sums. Then, these sums are added to the previous Energies 2020, 13, 3778 11 of 19

values kp0 and ki0 of PI parameter, respectively, so we can obtain the current PI parameters kp and ki as shown in Equation (5). ( kp = kp0 + ∆kp1 + ∆kp2 Energies 2020, 13, x FOR PEER REVIEW 11 of(5) 19 ki = ki0 + ∆ki1 + ∆ki2

4. Analysis Analysis of of Experimental Experimental Results Results

4.1. Experiment Experiment for for Sudden Sudden Change Change of of the the Load Load The 16 kWkW reversiblereversible three-phase three-phase PWM PWM converter converter was was built built as the as front-endthe front- converterend converter of the of motor the motordriver driver power power supply, supply, as shown as shown in Figure in 8Figure. The experimental8. The experimental prototype prototype consists consists of AC sideof AC board, side board,DC side DC board, side board, control control board, board, bus capacitor bus capacitor board board and bridge and bridge circuit circuit board. board. NPC-I NPC type-I three-phase type three- phasethree-level three PWM-level bridgePWM structure, bridge structure as shown, as in shown Figure 2 in, was Figure chosen 2, was for the chosen bridge for circuit the bridge board, circuit a DSP board,TMS320F28335 a DSP TMS320F28335 was employed was from employed TI Company from TI (Texas Company Instruments, (Texas Instruments, Inc, Dallas, TX,Inc, USA) Dallas, in TX, the USA)control in board, the control and FZ06NIA075SA board, and FZ06NIA075SA was chosen from was Vincotech chosen Company from Vincotech (Munich, Company Bayern, Germany)(Munich, Bayernfor the, three-phase Germany) for three-level the three PWM-phase bridge three- circuit.level PWM In this bridge experimental circuit. In prototype, this experimental the PWM prototype, converter theadopts PWM the converter double closed-loop adopts the control double strategy closed- basedloop control on the ACstrategy side voltagebased on vector the AC oriented side voltage control, vectoras shown oriented in Figure control3. , as shown in Figure 3.

Bus capacitor Control board board DC side board Bridge circuit board AC side board

FigureFigure 8. 8. ExperimentalExperimental prototype prototype of of 1 166 k kWW reversible reversible three three-phase-phase three three-level-level PWM converter converter..

The parameters of of the experimental prototype prototype are are as as follows: follows: the the AC AC side side inductance inductance is is 1 1 m mH,H, thethe switching frequencyfrequency isis 2020 kHz, kHz, and and the the AC AC voltage voltage frequency frequency is 50is 5 Hz.0 H Thez. The DC DC side side voltage voltage of theof theprototype prototype supply supply the back-endthe back-end PWM PWM converter converter of the of motor the motor driver driver power power supply, supply, and the and load the of load the ofdriver the driver power power supply supply is servo is motor. servo Accordingmotor. According to the law to the of conservation law of conservation of energy of in energy the dynamic in the dynamicprocess, the process, response the timeresponseTr of thetime voltage Tr of the outer voltage loop andouter the loop maximum and the variation maximum of the variation power ∆ofP themax powerare considered ΔPmax are in considered the design in of theDC designside capacitance. of DC sideThe capacitance. change of The energy change∆W ofon energy capacitance ΔW on is capacitancecalculated by is Equationcalculated (6). by Equation (6). ∆PmaxTr ∆W=PTmax r (6) W 2 (6) 2 The variation of DC side voltage ∆Udc is as follows: The variation of DC side voltage ΔUdc is as follows: ∆W ∆Udc =W (7) Udc CdUdc CU (7) d dc The DC voltage fluctuation in Equation (7) should be less than the maximum allowable value ΔUdcmax, so Equation (8) is obtained: TP C  r max d 2UU (8) dcmax dc When ΔPmax is set as 10% of the rated power, the voltage loop response time Tr is 200 ms and bus voltage fluctuation is 5%. The DC side capacitance can be calculated by Equation (8). Through calculation and comprehensive consideration, the design result of DC side capacitance is 2950 uF.

Energies 2020, 13, 3778 12 of 19

The DC voltage ﬂuctuation in Equation (7) should be less than the maximum allowable value ∆Udcmax, so Equation (8) is obtained:

Tr∆Pmax Cd (8) ≥ 2∆UdcmaxUdc

When ∆Pmax is set as 10% of the rated power, the voltage loop response time Tr is 200 ms and bus voltage ﬂuctuation is 5%. The DC side capacitance can be calculated by Equation (8). Through calculation and comprehensive consideration, the design result of DC side capacitance is 2950 uF. The rated operating condition of the front-end PWM converter is 220 V AC side phase voltage and 700 V DC bus voltage. Under the rated condition, the experiment of sudden loading and unloading is carried out for the reversible three-phase PWM converter with traditional PI controller and single fuzzy PI controller for the voltage outer loop respectively. The waveforms are shown in Figure9. In Figure9, CH1 is A-phase current waveform isa, CH2 is A-phase voltage waveform usa, CH3 is C-phase current waveform isc, and CH4 is DC bus voltage Udc waveform. Figure9a shows the waveforms using traditional PI controller for voltage outer loop, and Figure9d shows the waveform using traditional single fuzzy PI controller for voltage outer loop. Figure9b,c,e,f shows the details of the waveforms when the load changes. For these two kinds of voltage controller, when sudden change of the load is from null load to half load, DC bus voltage was dropped and can be restored to the given voltage value after a short period of adjustment; while sudden change of the load is from half load to null load, the DC bus voltage overshot and can be restored to the given DC voltage value after a short period of adjustment. The experimental data of traditional PI controller and single fuzzy PI controller used in the voltage outer loop are shown in Table7. From the Figure9 and Table7, it can be seen that the reversible three-phase PWM converter with traditional single fuzzy PI controller used in the voltage outer loop has smaller overshoot and drop, and faster dynamic response speed than the traditional PI controller. In view of the wide AC side and wide DC side voltage application range, the corresponding maximum and minimum values of AC side voltage and DC side given voltage were selected respectively, and the experiments for sudden change of the load using double fuzzy PI controller and traditional single fuzzy PI controller respectively were compared. Under the operating condition that the AC phase voltage is 200 V and the DC bus voltage is 650 V, the experiment of sudden loading and unloading is carried out for the reversible three-phase PWM converter with traditional single fuzzy PI controller and double fuzzy PI controller for the voltage outer loop respectively. The waveforms are shown in Figure 10. Under the operating condition that the AC phase voltage is 240 V and the DC bus voltage is 750 V, the same experiments were carried out. The waveforms are shown in Figure 11. The channels of the same physical quantity in Figures 10 and 11 are the same as above. The experimental data of traditional single fuzzy PI controller and double fuzzy PI controller used in the voltage outer loop are shown in Table8. From the Figures 10 and 11 and Table8, it can be seen that the reversible three-phase PWM converter with double fuzzy PI controller has smaller overshoot and drop and faster dynamic response speed than that with traditional single fuzzy PI controller. It can be seen that the performance using double fuzzy PI controller is much better than that using traditional single fuzzy PI controller when the AC voltage and DC given voltage changed in large range. The results of the experiment verify the eﬀectiveness of the proposed double fuzzy PI controller in this paper. Energies 2020, 13, x FOR PEER REVIEW 12 of 19

The rated operating condition of the front-end PWM converter is 220 V AC side phase voltage and 700 V DC bus voltage. Under the rated condition, the experiment of sudden loading and unloading is carried out for the reversible three-phase PWM converter with traditional PI controller and single fuzzy PI controller for the voltage outer loop respectively. The waveforms are shown in Figure 9. In Figure 9, CH1 is A-phase current waveform isa, CH2 is A-phase voltage waveform usa, CH3 is C-phase current waveform isc, and CH4 is DC bus voltage Udc waveform. Figure 9a shows the waveforms using traditional PI controller for voltage outer loop, and Figure 9d shows the waveform using traditional single fuzzy PI controller for voltage outer loop. Figure 9b,c,e,f shows the details of the waveforms when the load changes. For these two kinds of voltage controller, when sudden change of the load is from null load to half load, DC bus voltage was dropped and can be restored to the given voltage value after a short period of adjustment; while sudden change of the load is from half load to null load, the DC bus voltage overshot and can be restored to the given DC voltage value after a short period of adjustment. The experimental data of traditional PI controller and single fuzzy PI controller used in the voltage outer loop are shown in Table 7. From the Figure 9 and Table 7, it can be seen that the reversible three-phase PWM converter with traditional single fuzzy PI controller used in the voltage outer loop has smaller overshoot and drop, and faster dynamic response speed than the traditional PI controller. In view of the wide AC side and wide DC side voltage application range, the corresponding maximum and minimum values of AC side voltage and DC side given voltage were selected respectively, and the experiments for sudden change of the load using double fuzzy PI controller and traditional single fuzzy PI controller respectively were compared. Under the operating condition that the AC phase voltage is 200 V and the DC bus voltage is 650 V, the experiment of sudden loading and unloading is carried out for the reversible three-phase PWM converter with traditional single fuzzy PI controller and double fuzzy PI controller for the voltage outer loop respectively. The waveforms are shown in Figure 10. Under the operating condition that the AC phase voltage is 240 V and the DC bus voltage is 750 V, the same experiments were carried out. The waveforms are shown in Figure 11. The channels of the same physical quantity in Figure 10 and Figure 11 are the same as above. The experimental data of traditional single fuzzy PI controller and double fuzzy PI controller used in the voltage outer loop are shown in Table 8. From the Figures 10 and 11 and Table 8, it can be seen that the reversible three-phase PWM converter with double fuzzy PI controller has smaller overshoot and drop and faster dynamic response speed than that with traditional single fuzzy PI controller. It can be seen that the performance using double fuzzy PI controller is much better than that using traditional single fuzzy PI controller when the AC voltage and DC given voltage changed in large range. The results of the experiment verify the effectiveness of the proposed double fuzzy PI Energiescontroller2020, 13, 3778 in this paper. 13 of 19

CH2:usa(1000V/div)

CH4:Udc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

Energies 2020, 13, x FOR PEER REVIEW 13 of 19 (a)

CH2:usa CH2:usa

CH4:Udc CH3:isc CH4:Udc CH3:isc

CH1:i CH1:isa sa (b) (c)

CH2:usa(1000V/div)

CH4:Udc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

(d)

CH2:usa CH2:usa

CH4:Udc CH4:Udc CH3:isc CH3:isc

CH1:i sa CH1:i sa

(e) (f)

FigureFigure 9. Waveforms 9. Waveforms adopting adopting traditional traditional PI controller controller and and single single fuzzy fuzzy PI controller PI controller respectively respectively for for voltagevoltage outer outer loop loop (AC: (AC: 220 22 V,0 V DC:, DC: 700 700 V).V). ( a)) Waveforms Waveforms adopting adopting traditional traditional PI controller PI controller;; (b) zoom (b ) zoom in ofin waveform of waveform under under sudden sudden load load (traditional (traditional PI PI controller controller);); (c) (zcoom) zoom in of in w ofaveform waveform under undersudden sudden unload (traditional PI controller); (d) waveforms adopting single fuzzy PI controller; (e) zoom in of unload (traditional PI controller); (d) waveforms adopting single fuzzy PI controller; (e) zoom in of waveforms under sudden load (single fuzzy PI controller); (f) zoom in of waveforms under sudden waveforms under sudden load (single fuzzy PI controller); (f) zoom in of waveforms under sudden unload (single fuzzy PI controller). unload (single fuzzy PI controller).

Energies 2020, 13, x FOR PEER REVIEW 14 of 19 Energies 2020, 13, 3778 14 of 19 Energies 2020, 13, x FOR PEER REVIEW 14 of 19

CH2:usa(1000V/div)

CH4:Udc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

(a) (a) CH2:usa(1000V/div)

CH2:usa(1000V/div)

CH4:Udc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

(b) (b) Figure 10. Waveforms adopting traditional single fuzzy PI controller and double fuzzy PI controller respectivelyFigure 10. Waveforms for voltage adopting outer loop traditional (AC: 200 single V, DC: fuzzy 650 V PI) . ( acontroller ) Waveforms and adoptingdouble fuzzy traditional PI control controller singleler fuzzyrespectively PI controller for voltage ; (b) w outer aveforms loop (AC:adopting 20 2000 V V,double, DC: 65065 fuzzy0 V V).). PI( a )control Waveforms ler. adopting traditional single fuzzy PI controller;controller; (b) waveformswaveforms adopting double fuzzyfuzzy PIPI controller.controller.

CH2:usa(1000V/div) CH2:usa(1000V/div)

CH4:Udc(100V/div) CH4:Udc(100V/div) CH1:isa(20A/div) CH1:isa(20A/div)

CH3:isc(20A/div) CH3:isc(20A/div)

(a) (a) Figure 11. Cont.

Energies 2020, 13, 3778 15 of 19 Energies 2020, 13, x FOR PEER REVIEW 15 of 19

CH2:usa(1000V/div)

CH4:Udc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

(b)

Figure 11. Waveforms adopting traditional single fuzzy PI controller and double fuzzy PI controllercontroller respectively for voltage outer loop (AC: 24 2400 V V,, DC: 750750 V V).). ( a) Waveforms adopting traditional single fuzzy PI controller;controller; (b) waveformswaveforms adopting double fuzzyfuzzy PIPI controller.controller. Table 7. Comparison of experimental data with traditional PI controller and single fuzzy PI controller,Table 7. Comparison respectively. of experimental data with traditional PI controller and single fuzzy PI controller, respectively. Traditional PI Single Fuzzy PI AC 220 V, DC 700 V Traditional PI Single Fuzzy PI AC 220 V, DC 700 V Controller Controller Controller Controller Voltage drop 32 V 16 V FromFrom null null load load to to half half load Voltage drop 32 V 16 V Adjustment time 280 ms 160 ms load Adjustment time 280 ms 160 ms VoltageVoltage overshoot 30 V 12 V FromFrom half half load load to to null null load 30 V 12 V overshootAdjustment time 310 ms 170 ms load Adjustment time 310 ms 170 ms Table 8. Comparison of experimental data with single fuzzy PI controller and double fuzzy PI controller,Table 8. Comparison respectively. of experimental data with single fuzzy PI controller and double fuzzy PI controller, respectively. Traditional Single Double Fuzzy PI AC 200 V, DC 650 V TraditionalFuzzy Single PI ControllerFuzzy PI DoubleController Fuzzy PI AC 200 V, DC 650 V Voltage dropController 34 VController 20 V From null load to half load Voltage dropAdjustment time34 V 240 ms 1002 ms0 V From null load to Adjustment half load Voltage overshoot240 ms 39 V 1910 V0 ms From half load to null load time Adjustment time 210 ms 110 ms Voltage Traditional39 V Single Double Fuzzy19 V PI From half load toAC 240 V,overshoot DC 750 V Fuzzy PI Controller Controller null load Adjustment Voltage drop210 ms 48 V 2811 V0 ms From null load to half load time AdjustmentT timeraditional Single 260 F msuzzy PI Double 120 ms Fuzzy PI AC 240 V, DC 750 V Voltage overshootController 33 VC 18ontrol V ler From half load to null load Voltage dropAdjustment time48 V 290 ms 1502 ms8 V From null load to Adjustment half load 260 ms 120 ms The selection of DC side capacitancetime should be considered. On the one hand, the larger the value Voltage of DC side capacitance, the better the filtering effect; on the33 other V hand, considering the volume,18 V weight, price,From and half dynamic load to characteristics, overshoot the value of DC side capacitance should not be too large. Therefore, null load Adjustment two capacitances with different values are used for comparison.290 ms Figure 11b shows waveforms150 ms adopting double fuzzy PI controller withtime the 2950 µF DC side capacitance. Figure 12 shows waveforms adopting double fuzzy PI controller with the 1600 µF DC side capacitance. The experimental data of adopting The selection of DC side capacitance should be considered. On the one hand, the larger the value of DC side capacitance, the better the filtering effect; on the other hand, considering the volume,

Energies 2020, 13, x FOR PEER REVIEW 16 of 19

Energiesweight, 20 price20, 13, xand FOR dynamic PEER REVIEW characteristics, the value of DC side capacitance should not be too16 large. of 19 EnergiesTherefore,2020, 13, 3778 two capacitances with different values are used for comparison. Figure 11b shows16 of 19 weight,waveforms price adopting, and dynamic double characteristics, fuzzy PI control the lervalue with of theDC 2950side capacitanceμF DC side should capacitance not be. Figuretoo large. 12 Therefore,shows waveforms two capacitance adoptings double with differentfuzzy PI valuecontrols ler are with used the for 1600 comparison μF DC side. Figure capacitance 11b shows. The 2950wexperimentalµaveformsF and 1600 adopting dataµF DCof doubleado sidepting capacitancesfuzzy 2950 PI μF control and with 1600ler withdouble μF DCthe fuzzy2950side capacitanceμF PI DC controller sides capacitance with are double shown. Figure fuzzy in Table12 PI 9. Figureshowscontrol 11b andwleraveforms are Table shown9 showadopting in Table that double 9 using. Figure fuzzy smaller 11b PIand DCcontrol Table sideler 9 capacitance show with thatthe 1600using can μF smaller make DC side the DC dynamic capacitanceside capacitance response. The of the systemexperimentalcan make faster. the dynamic data of ado responsepting 2950of the μF system and 1600faster μF. DC side capacitances with double fuzzy PI controller are shown in Table 9. Figure 11b and Table 9 show that using smaller DC side capacitance can make the dynamicCH2: responseusa(1000V/div) of the system faster.

CH2:usa(1000V/div)

CH4:Udc(100V/div) CH1:i (20A/div) CH4:Usadc(100V/div)

CH1:isa(20A/div)

CH3:isc(20A/div)

CH3:isc(20A/div) Figure 12. Waveforms adopting double fuzzy PI controller with the 1600 μF DC side capacitance. Figure 12. Waveforms adopting double fuzzy PI controller with the 1600 µF DC side capacitance. TableTable 9.FigureComparison 9. 12.Comparison Waveforms of experimentalof adoptingexperimental double data data fuzzy ofof adoptingadopting PI control d doubleoubleler with fuzzy fuzzy the PI1600 PIcontrol μF controller DCler side with withcapacitance different diﬀerent DC. DC side capacitances.side capacitances. Table 9. Comparison of experimental data of adopting double fuzzy PI controller with different DC DC Side Capacitance side capacitances. AC 240 V, DC 750 V DC Side Capacitance AC 240 V, DC 750 V 2950 μF 1600 μF µ µ Voltage drop 2950DC2 8Side VF Capacitance26 V 1600 F From null loadAC to 24 half0 V load, DC 750 V VoltageAdjustment drop time 2950 28120 V m μFs 160090 m μFs 26 V From null load to half load Voltage drop 28 V 26 V From null load to half loadAdjustment Voltage timeovershoot 12018 ms V 15 V 90 ms From half load to null load Adjustment time 120 ms 90 ms VoltageAdjustment overshoot time 18150 V ms 110 m 15s V From half load to null load Voltage overshoot 18 V 15 V From half load to null loadAdjustment time 150 ms 110 ms 4.2. Practical Application of Double Fuzzy PI ControllerAdjustment time 150 ms 110 ms The experimental prototype is used as the front-end PWM converter of the motor driver power 4.2. Practical4.2. Practical Application Application of Doubleof Double Fuzzy Fuzzy PI PI ControllerController supply, and the voltage outer loop of experimental prototype adopts the proposed double fuzzy PI Thecontroller.The experimental experimental The sinusoidal prototype prototype speed is issignal used used asis as given thethe front-endfront to the-end motor PWM PWM load converter converter of the ofdriver the of themotor power motor driver supply, driver power the power supply,supplyperiod and ,of and the given voltagethe sinusoidalvoltage outer outer speed loop loop signal of of experimental experimental is 5 s, and itsprototype prototype peak value adopts adopts corresponds the the proposed proposed to the double given double fuzzyvalue fuzzy PIof PI controller.controller.the maximum The sinusoidalThe speed. sinusoidal The speed waveformsspeed signal signal is are given is shown given to theinto Figurethe motor motor 1 load3. load of theof the driver driver power power supply, supply, the the period period of given sinusoidal speed signal is 5 s, and its peak value corresponds to the given value of of given sinusoidal speed signal is 5 s, and its peak value corresponds to the given value of the the maximum speed. The waveforms are shown in Figure 13. maximum speed. The waveforms are shown in Figure 13.

CH1:isa(10A/div) CH3:isc(10A/div) CH2:usa(1000V/div)

CH4:us(10V/div)

1.62s 0.88s 1.62s 0.88s CH4:us(10V/div) rectifying inverting rectifying inverting 1.62sMotor 0.88s 1.62sMotor 0.88s t(1s/div) rectifforwardying inverting rectifreversalying inverting Motor Motor t(1s/div) forward reversal

Figure 13. Experimental waveforms with sinusoidal speed given signal.

In Figure 13, Channels CH1 and CH3 are A-phase current and C-phase current waveforms, respectively; CH2 is A-phase voltage waveform; and CH4 is the sinusoidal speed given signal. Observe the waveform of a sinusoidal cycle. The converter works in the rectiﬁer state ﬁrst. At this Energies 2020, 13, 3778 17 of 19 time, the motor rotates forward and accelerates with the given sinusoidal speed signal. Due to the existence of rotation inertia, the motor starts to decelerate at 1.62 s (instead of 1.25 s corresponding to the peak of voltage signal). At this time, the converter switches to the inverter state and feeds back the energy generated by motor decelerating to the AC side. When the motor reverses, its operating principle is the same as that of forward rotation. It can be seen that the proposed double fuzzy PI controller can achieve good performance when used in a reversible PWM converter with a wide range of AC side voltage and DC side voltage.

5. Conclusions The double fuzzy PI controller for the voltage outer loop of the reversible three-phase PWM converter is proposed in this paper. This method not only does not depend on the precise model of the control object but can also adjust the PI control parameters adaptively, which is equivalent to changing the static operating point of the control object under speciﬁc operating conditions, making the system adaptive to a new balance point. On the basis of the traditional fuzzy PI controller, the double fuzzy PI controller is realized by adding the sub fuzzy controller, which generates the auxiliary correction of PI parameters due to the AC side voltage variation and DC side given voltage variation of the reversible three-phase PWM converter. Therefore, the proposed double fuzzy PI controller can realize the stable control of the reversible three-phase PWM converter with wide AC side voltage and DC side voltage, and obtain fast dynamic response performance. In future work, we will apply the design idea of the double fuzzy controller in this paper to the control of the back-end inverter of the motor driver power supply in order to realize the universality of the motor driver.

Author Contributions: Conceptualization, W.Z. and Y.F.; Methodology, W.Z.; Software and investigation, R.Y. and Y.F.; Validation, Z.W. and W.Z.; Formal analysis, Y.F. and Z.W.; Writing—review and editing, W.Z. and R.Y. All authors have read and agreed to the published version of the manuscript. Funding: Supported by the Equipment Pre-research Project of China (No.31512040201-2) and supported by the Science and Technology Cooperation Fund of Yangzhou City Hall project (No. YZ2018136). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

PWM Pulse Width Modulation PI Proportional Integral IGBT Insulated Gate Bipolar Transistor NPC Neutral Point Clamped SVPWM Space Vector Pulse Width Modulation PLL Phase-Locked Loop I-Park Inverse Park Transformation

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