Predictive Control of a Three-Phase Neutral Point Clamped Inverter

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Finally, each capacitor from the DC-link fulfills the followingdynamic equation:V c (k +1)=V c (k)+ 1 C i c(k)T s (9)where i c (k) is the current through the capacitor, v c (k) isits voltage and C is the capacitance. Using (9), is possibleto obtain predictions for the future value of the capacitorsvoltage, based on its present current and voltage.III. PWM CURRENT CONTROL METHODBefore exposing the proposed predictive control method,a short review of classic PWM current control applied to athree-phase neutral point clamped inverter is presented, inorder to obtain suitable comparisons. The PWM scheme isshown in Fig. 3. The load current is measured and comparedFig. 2. Possible voltage vectors and switching states generated by a threelevelinverter.i*(k)i(k)v*ababcS aSPulse Width bModulationS c+ -Fig. 3. Classic PWM Current Control Method.3 M3fwhere a = e j 2π 3 .Applying a sampling period T s , the derivative formdi(t)/dt is approximated by:di(t) i(k) − i(k − 1)≈ (5)dt T sReplacing (5) in (1) and shifting the discrete time one stepforward, the relation between the discrete-time variables canbe described as:[T s Li(k +1)=i(k)+v(k +1)− e(k +1)](6)RT s + L T sEquation (6) is used to obtain predictions for the future valueof the load current, i(k +1), considering all possible voltagevectors v generated by the inverter and measured current atthe k th sampling interval. The control strategy also uses anestimation of the future reference current, which is obtainedby a second order extrapolation given by:i ∗ (k +1)=3i ∗ (k) − 3i ∗ (k − 1) + i ∗ (k − 2) (7)The current prediction from (6) requires also the future loadback-EMF, e(k +1). That value can be estimated usinga second order extrapolation, analog to the one appliedto compute the future reference current. Present and pastestimations of e, needed for the extrapolation, can be obtainedfrom the load equation (6) shifted backward in time, and loadcurrent measurements as:ê(k) =v(k)+ L i(k − 1) − RT s + Li(k) (8)T s T sFor a sufficiently small sampling time, it is possible toconsider i ∗ (k +1) ≈ i ∗ (k) and e(k +1) ≈ e(k), sonoextrapolation is needed.with its reference value. Next, a proportional-integral (PI)controller generates the reference load voltages that enter amodulator, where each voltage is compared with two triangularcarrier signals (superior and inferior). The switching stateapplied to the inverter is selected according to the results ofthe comparisons. For more details, see [2].IV. PREDICTIVE CURRENT CONTROL METHODFig. 4 shows a scheme that summarizes the implementedcontrol strategy. The future value of the load current andvoltages in the capacitors are predicted for the 27 switchingstates generated by the inverter, by means of equations (6)and (9). For this purpose, it is necessary to measure thepresent load current and voltages in the capacitors. Afterobtaining the predictions, a quality function g is evaluatedfor each switching state. The switching state that minimizesg is selected and applied during the next sampling period. Theproposed quality function has the following composition:g = f(i ∗ (k +1), i(k + 1)) + λ · h(Vc 12 (k +1),n c ) (10)where n c is the number of commutations to get to theswitching state under evaluation. The first term in (10),f(i ∗ , i), is dedicated to achieve reference tracking, quantifyingthe difference between the reference current and currentprediction on the next sampling time, for a given switchingi(k)1365
i*(k+1)(i(k+1),Vc 1,2 (k+1)) i27S aMinimization Sof bg function S cPredictivemodeli(k)Vc 1,2 (k)3 M3fTABLE ICIRCUIT PARAMETERS.C 1 , C 2 1[mF]R 0.5[Ω]L 10[mH]EMF Amplitude 50[V]EMF Frequency 50[Hz]Fig. 4.Predictive Current Control Method.(a)PWMstate. The following composition of f, or”tracking cost”, isproposed:f(i ∗ (k +1), i(k + 1)) = |i ∗ α(k +1)− i α (k +1)|+ |i ∗ β (k +1)− i β(k +1)| (11)where i α and i β are the real and imaginary components ofcurrent vector i, respectively, and i ∗ α and i ∗ βare the real andimaginary components of the reference current vector i ∗ .The objective of the second term in (10), λ · h(Vc 12 ,n c ),is to take advantage of the state redundancy of a three-levelinverter from the fact that the tracking cost f depends onlyon the voltage vector selected. Two possible compositions ofh are proposed:{ = |Vc1 (k +1)− Vch(Vc 12 (k +1),n c )2 (k +1)|(12)= n cThe first choice, adds up to g a term proportional to theabsolute difference between both capacitors voltage predictions.A switching state that generates smaller differenceswill be preferred. The second option is simply the number ofcommutations to get to the next switching state. A switchingstate that implies fewer commutations will be preferred. Theλ weighting factor multiplying h in (10), handles the relationbetween f and h in g. A small λ implies greater priority toreference tracking over balance in the DC-link or reductionof the switching frequency.V. RESULTSResults for the proposed predictive current control strategyare presented. Within this section, the DC-link will be maintainedin 200[V ], following a previous rectification stage. Thesampling period applied is T s = 100[µs]. Values of the DC-Link and load parameters (Fig. 1) are shown in Table I.A. Reference Tracking.Performance of the proposed strategy is analyzed andcompared with PWM current control. The algorithm wasimplemented using the following quality function:g = |i ∗ α(k +1)− i α (k +1)| + |i ∗ β(k +1)− i β (k +1)| (13)In order to generate the same average switching frequency persemiconductor as the predictive method, of about 835[Hz],the PWM carrier frequency was set to 1670[Hz].i α,β[A]i α,β[A]20100−10−2020100−10−200 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04(b)Predictive0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Time [s]Fig. 5.Load current response : (a) PWM (b) Predictive.Waveforms obtained for the load current, load voltage andvoltage spectrum for both methods are presented in Figs. 5,6 and 7. Figs. 5 and 6 show the initial transient response,starting from zero load current until reaching stationarystate. The load voltage spectrum obtained with the Predictivemethod is more spread than with the PWM method, as shownin Fig. 7. This could be considered a disadvantage of themethod.In order to observe the interaction between both componentsof the load current, the amplitude of i ∗ α (real componentof the reference current) is reduced from 20[A] to 10[A] attime t =0.015[s]. The imaginary component is left at 20[A].Results for the load current are presented in Figs. 8 and 9, forPWM and Predictive methods. From the presented results,it is clear that the predictive method achieves comparableperformance on reference tracking during transient response,without needing modulation techniques or to adjust any kindof linear controllers. In addition, note that the proposedmethod presents no interaction between i α and i β . Thedecoupling of both components of the current vector is aninherent property of the predictive method.B. Balance in the DC-link.Balance is achieved including the term presented in alternative1 of equation (12). The method was implemented1366
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Predictive Control of a Three-Phase Neutral Point Clamped Inverter

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