Predictive Control of a Three-Phase Neutral-Point-Clamped Inverter

Predictive Control of a Three-Phase Neutral Point Clamped Inverter J. Rodr´ıguez, J. Pontt, P. Cort´es, R. Vargas Universidad T´ecnica Federico Santa Mar´ıa Department of Electronic Engineering Avenida Espa˜na 1680, Casilla 110-V, Valpara´ıso, Chile E-mail: [email protected] - http://www.elo.utfsm.cl Abstract— A new predictive strategy for current control of a II. M ODEL OF THE S YSTEM three-phase neutral point clamped inverter is presented. It is based on a discrete-time model of the system, used to predict Fig. 1 shows a model of the system. It includes a three- future values of the load current and voltage of the capacitors in phase three-level inverter and a resistive-inductive-active the DC-link, for each possible switching state generated by the load. The converter applies to the load 19 voltage vectors, inverter. The state that minimizes a given quality function ”g” is generated from 27 switching states, as presented in Fig. 2. selected to be applied during the next sampling interval. Several compositions of g are proposed, including terms dedicated to achieve reference tracking, balance in the DC-link and reduction of the switching frequency. The algorithm uses the redundancy of switching states, typical of a three-level inverter, by means of a simple strategy. In comparison with classic PWM current control, the strategy presents a remarkable performance. The proposed method achieves comparable reference tracking with lower switching frequency per semiconductor and a slightly improved transitory behavior. It requires a greater sampling frequency, which should not be a problem, considering the present technologies available in digital signal processors. The main advantage of the method is that it does not require any kind of linear controller or modulation technique, achieving a different approach to control a power converter. I. I NTRODUCTION Current control in three-phase inverters has been exten- sively studied. Between the most common methods, litera- ture states non-linear techniques, like hysteresis control, and linear methods, like the use of a PI controller in conjunction with pulse wide modulation (PWM) [1],[2]. On the other Fig. 1. Circuit of a three-phase neutral point clamped inverter connected hand, three-level neutral point clamped inverters have earned to a resistive-inductive-active load. popularity between medium voltage drives [3],[4],[5]. Model Predictive Control (MPC) is a control theory which was developed at the end of the Seventies [6]. Variants of The center of a predictive control algorithm is the model this type of control strategy have found application in power of the plant, from which predictions are obtained. In this converters. Predictive control has been used in drives [7], case, it corresponds to the equation of a three-phase resistive- power factor correction [8] and active filters [9]. All of these inductive-active load, which fulfills: works consider linear models and use modulation techniques di(t) L = v(t) − Ri(t) − e(t) (1) for voltage generation. In [10] and [11], a new variant of dt MPC is used to control a matrix converter and a three-phase where R and L are the load resistance and inductance two-level inverter, respectively. In both cases, the idea is to respectively, v is the voltage vector generated by the inverter, apply, from the converter, the switching state that minimizes e is the electromotive force of the load and i is the load a quality function evaluated for the next sampling time. In current vector. These vectors are defined as: this document, a similar technique is applied, achieving a 2 new control method for a three-phase neutral point clamped v = (Va0 + aVb0 + a2 Vc0 ) (2) 3 inverter. Several variations of the algorithm are studied and 2 compared with classic PWM control, including features like i = (ia + aib + a2 ic ) (3) 3 reference tracking, balance in the DC-link and reduction of 2 the switching frequency. e = (ea + aeb + a2 ec ) (4) 3 0-7803-9033-4/05/$20.00 ©2005 IEEE. 1364  Finally, each capacitor from the DC-link fulfills the fol- lowing dynamic equation: 1 Vc (k + 1) = Vc (k) + ic (k)Ts (9) C where ic (k) is the current through the capacitor, vc (k) is its voltage and C is the capacitance. Using (9), is possible to obtain predictions for the future value of the capacitors voltage, based on its present current and voltage. III. PWM C URRENT C ONTROL M ETHOD Before exposing the proposed predictive control method, a short review of classic PWM current control applied to a three-phase neutral point clamped inverter is presented, in order to obtain suitable comparisons. The PWM scheme is shown in Fig. 3. The load current is measured and compared Sa i*(k) v* a a Sb 3 Fig. 2. Possible voltage vectors and switching states generated by a three- b Pulse Width M +- b Modulation Sc 3f level inverter. c i(k) 2π where a = ej 3 . Applying a sampling period Ts , the derivative form i(k) di(t)/dt is approximated by: di(t) i(k) − i(k − 1) Fig. 3. Classic PWM Current Control Method. ≈ (5) dt Ts Replacing (5) in (1) and shifting the discrete time one step with its reference value. Next, a proportional-integral (PI) forward, the relation between the discrete-time variables can controller generates the reference load voltages that enter a be described as: modulator, where each voltage is compared with two triangu-
 Ts L lar carrier signals (superior and inferior). The switching state i(k + 1) = i(k) + v(k + 1) − e(k + 1) (6) RTs + L Ts applied to the inverter is selected according to the results of Equation (6) is used to obtain predictions for the future value the comparisons. For more details, see [2]. of the load current, i(k + 1), considering all possible voltage vectors v generated by the inverter and measured current at IV. P REDICTIVE C URRENT C ONTROL M ETHOD the k th sampling interval. The control strategy also uses an Fig. 4 shows a scheme that summarizes the implemented estimation of the future reference current, which is obtained control strategy. The future value of the load current and by a second order extrapolation given by: voltages in the capacitors are predicted for the 27 switching states generated by the inverter, by means of equations (6) i∗ (k + 1) = 3i∗ (k) − 3i∗ (k − 1) + i∗ (k − 2) (7) and (9). For this purpose, it is necessary to measure the The current prediction from (6) requires also the future load present load current and voltages in the capacitors. After back-EMF, e(k + 1). That value can be estimated using obtaining the predictions, a quality function g is evaluated a second order extrapolation, analog to the one applied for each switching state. The switching state that minimizes to compute the future reference current. Present and past g is selected and applied during the next sampling period. The estimations of e, needed for the extrapolation, can be obtained proposed quality function has the following composition: from the load equation (6) shifted backward in time, and load current measurements as: g = f (i∗ (k + 1), i(k + 1)) + λ · h(V c12 (k + 1), nc ) (10) L RTs + L where nc is the number of commutations to get to the ˆe(k) = v(k) + i(k − 1) − i(k) (8) Ts Ts switching state under evaluation. The first term in (10), For a sufficiently small sampling time, it is possible to f (i∗ , i), is dedicated to achieve reference tracking, quantify- consider i∗ (k + 1) ≈ i∗ (k) and e(k + 1) ≈ e(k), so no ing the difference between the reference current and current extrapolation is needed. prediction on the next sampling time, for a given switching 1365  TABLE I i*(k+1) Sa C IRCUIT PARAMETERS . Minimization Sb 3 of M g function Sc 3f (i(k+1),Vc1,2(k+1))i C 1 , C2 1[mF] 27 R 0.5[Ω] i(k) L 10[mH] Predictive EMF Amplitude 50[V] model Vc1,2(k) EMF Frequency 50[Hz] Fig. 4. Predictive Current Control Method. (a) PWM 20 state. The following composition of f , or ”tracking cost”, is 10 iα,β [A] proposed: 0 −10 f (i∗ (k + 1), i(k + 1)) = |i∗α (k + 1) − iα (k + 1)| −20 + |i∗β (k + 1) − iβ (k + 1)| (11) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 (b) Predictive where iα and iβ are the real and imaginary components of 20 current vector i, respectively, and i∗α and i∗β are the real and 10 imaginary components of the reference current vector i∗ . iα,β [A] 0 The objective of the second term in (10), λ · h(V c12 , nc ), −10 is to take advantage of the state redundancy of a three-level −20 inverter from the fact that the tracking cost f depends only 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time [s] on the voltage vector selected. Two possible compositions of h are proposed: Fig. 5. Load current response : (a) PWM (b) Predictive.  = |V c1 (k + 1) − V c2 (k + 1)| h(V c12 (k + 1), nc ) (12) = nc The first choice, adds up to g a term proportional to the Waveforms obtained for the load current, load voltage and absolute difference between both capacitors voltage predic- voltage spectrum for both methods are presented in Figs. 5, tions. A switching state that generates smaller differences 6 and 7. Figs. 5 and 6 show the initial transient response, will be preferred. The second option is simply the number of starting from zero load current until reaching stationary commutations to get to the next switching state. A switching state. The load voltage spectrum obtained with the Predictive state that implies fewer commutations will be preferred. The method is more spread than with the PWM method, as shown λ weighting factor multiplying h in (10), handles the relation in Fig. 7. This could be considered a disadvantage of the between f and h in g. A small λ implies greater priority to method. reference tracking over balance in the DC-link or reduction In order to observe the interaction between both compo- of the switching frequency. nents of the load current, the amplitude of i∗α (real component V. R ESULTS of the reference current) is reduced from 20[A] to 10[A] at Results for the proposed predictive current control strategy time t = 0.015[s]. The imaginary component is left at 20[A]. are presented. Within this section, the DC-link will be main- Results for the load current are presented in Figs. 8 and 9, for tained in 200[V ], following a previous rectification stage. The PWM and Predictive methods. From the presented results, sampling period applied is Ts = 100[µs]. Values of the DC- it is clear that the predictive method achieves comparable Link and load parameters (Fig. 1) are shown in Table I. performance on reference tracking during transient response, without needing modulation techniques or to adjust any kind A. Reference Tracking. of linear controllers. In addition, note that the proposed Performance of the proposed strategy is analyzed and method presents no interaction between iα and iβ . The compared with PWM current control. The algorithm was decoupling of both components of the current vector is an implemented using the following quality function: inherent property of the predictive method. g = |i∗α (k + 1) − iα (k + 1)| + |i∗β (k + 1) − iβ (k + 1)| (13) B. Balance in the DC-link. In order to generate the same average switching frequency per semiconductor as the predictive method, of about 835[Hz], Balance is achieved including the term presented in al- the PWM carrier frequency was set to 1670[Hz]. ternative 1 of equation (12). The method was implemented 1366  (a) PWM PWM 150 100 20 50 10 VaN [V] iα [A] 0 0 −50 −10 −100 −20 −150 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 (b) Predictive 150 20 100 10 50 iβ [A] VaN [V] 0 0 −50 −10 −100 −20 −150 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time [s] Time [s] Fig. 8. Load current response applying step on i∗α : PWM Method. Fig. 6. Load voltage on phase a: (a) PWM (b) Predictive. Predictive (a) PWM 40 20 10 Magnitude [%] 30 i [A] 0 α 20 −10 10 −20 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0 2.0 4.0 6.0 8.0 10.0 (b) Predictive 20 40 10 Magnitude [%] i [A] 30 0 β 20 −10 −20 10 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 Time [s] 0 2.0 4.0 6.0 8.0 10.0 Frequency [kHz] Fig. 9. Load current response applying step on i∗α : Predictive Method. Fig. 7. Load voltage spectrum: (a) PWM (b) Predictive. The λ weighting factor was set at 0.001. The switching using the following quality function: frequency fs was reduced from 835[Hz] to 645[Hz] without harming reference tracking. Increasing λ to 0.332 (emphasis g = |i∗α (k + 1) − iα (k + 1)| + |i∗β (k + 1) − iβ (k + 1)| in reducing the switching frequency), a frequency of 299[Hz] +λ|V c1 (k + 1) − V c2 (k + 1)| (14) was achieved. That represents only a 35.8% of the original average switching frequency per semiconductor or the fs The λ weighting factor was selected to be sufficiently small to presented by the PWM method. Results for the load current allow the algorithm to select switching states within a given on phase a, for λ = 0.001 and λ = 0.332, are presented in voltage vector. The value selected was λ = 0.001. As shown Fig. 11. in Fig. 10, the presented method succeeded maintaining As expected, applying a greater λ implies in most cases voltage balance without affecting the current control. reducing the switching frequency. In general, the trade-off is a slight increase of the reference tracking error. In the C. Reduction of the switching frequency. case presented (Fig.11) the mean absolute reference tracking Choosing alternative 2 of equation (12), it is possible error increased from 0.3137[A] to 0.3534[A]. To expose the to reduce considerably the average switching frequency per possibilities of the proposed method, a graph showing the semiconductor, fs . This was proven using the following relation between the design parameter λ and the switching quality function: frequency and mean absolute reference tracking error is presented in Fig.12. g = |i∗α (k + 1) − iα (k + 1)| + |i∗β (k + 1) − iβ (k + 1)| The designer should select a λ parameter that fits his +λ · nc (15) requirements in terms of switching frequency and reference 1367  λ v/s Switching Frequency (a) Without applying balance 0.9 200 Frequency [kHz] 0.8 0.7 Switching Vc ,Vc [V] 150 0.6 2 100 0.5 0.4 1 50 0.3 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 (b) Applying balance λ v/s Absolute Error 200 0.8 0.6 Absolute Vc1,Vc2 [V] Error [A] 150 0.4 100 0.2 50 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 Time [s] λ Fig. 10. Voltage in the capacitors of the DC-link: (a) Without applying Fig. 12. Design parameter λ (a)Relation with the mean switching frequency balance, quality function (13) (b) Applying balance, quality function (14). per semiconductor (b)Relation with the mean absolute reference tracking error. (a) High switching frequency 20 It presents a very effective control of the load current, and 10 compares well with established control methods, like PWM, i [A] 0 achieving a slightly better dynamic response. The proposed a −10 method presents no interaction between both components of −20 the load current. 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 One of the remarkable aspects of the method is the use of (b) Low switching frequency state redundancy, in order to achieve balance in the DC-link 20 and reduction of the switching frequency. The strategy allows 10 the designer to adjust the λ parameter to fits his requirements i [A] 0 in terms of switching frequency and reference tracking. a −10 The method can be easily implemented taking advantage of −20 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 the present technologies available in digital signal processors. Time [s] This control strategy uses, in a very convenient way, the discrete nature of the power converter and the microprocessor Fig. 11. Load current on phase a (a)High switching frequency: fs = used in the control. 645[Hz] (b)Emphasis in reducing the switching frequency: fs = 299[Hz]. These results show that predictive control is a very power- ful tool, with a conceptually different approach, which opens new possibilities in the control of power converters. tracking. The lowest mean absolute reference tracking error ACKNOWLEDGMENT achieved was 0.2745[A] or 1.3725% with λ = 0.062 and fs = 662[Hz]. The PWM method presented a mean absolute The authors acknowledge the support of the Chilean Re- reference tracking error of 0.2939[A] or 1.4695% with fs = search Fund CONICYT (Grant 1030368) and of the Univer- 835[Hz]. sidad T´ecnica Federico Santa Mar´ıa. Summarizing, the reference tracking performance of the R EFERENCES predictive method compares well with PWM current con- [1] J. Holtz, Pulsewidth Modulation for Electronic Power Conversion, trol, with lower switching frequency per semiconductor and Proceedings of the IEEE, 82(8), 1994. improved transitory behavior. Nevertheless, the proposed [2] M.P. Kazmierkowski, R. Krishnan and F. Blaabjerg, Control in Power Electronics, Academic Press, 2002. method requires a greater sampling frequency or data acqui- [3] A. Nabae, I. Takahashi, H. Akagi, A New Neutral-Point-Clamped PWM sition frequency. The previous fact should not be a problem, Inverter, IEEE Transactions on Industry Applications, vol. 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