Comparison of Direct and PWM Model Predictive Control for Power ...

Comparison of Direct and PWM Model Predictive Control for Power Electronic and Drive Systems Matthias Preindl, Silverio Bolognani University of Padova, Department of Industrial Engineering Via Gradenigo 6/A, 35100 Padova, Italy [email protected]

Abstract—In this paper, we compare the performance of the different Model Predictive Control (MPC) approaches for power electronic and drive systems: Finite Control Set (FCS) MPC directly applies the voltage vectors and Continuous Control Set (CCS) MPC uses voltage modulation (PWM, SVM). A unified system formulation is proposed to design both FCSMPC and CCS-MPC for three-phase resistive-inductive loads with an internal voltage source. The controllers are implemented with the explicit MPC technique and they are evaluated with a permanent-magnet-synchronous-machine drive system. The steady-state performance is assessed in terms of current quality and switching frequency and the dynamic performance is evaluated estimating the control bandwidth. Although CCSMPC and FCS-MPC have different characteristics (harmonic spectrum, constant or variable switching frequency, THD, etc.), they achieve similar performance. However, CCS-MPC can work with larger sampling-times, i.e. it has lower requirements on the control architecture. This makes it more versatile for application on different systems and power levels.

I. I NTRODUCTION Model Predictive Control (MPC) has been introduced to power electronic and drive systems in the last decade [1], [2]. It has been shown to be a promising alternative [3]– [13] to todays most popular concepts; vector control [14] and direct torque control [15], [16]. MPC is designed for discrete systems. Each time step, a Constrained Finite Time Optimal Control (CFTOC) problem [17] with cost function and constraints is solved. The system dynamic is described by a state-space model and the optimization is carried out over N future time steps, called the prediction horizon. The CFTOC is generally solved with numerical solvers for finding, if possible, the feasible optimal solution. The CFTOC problem is solved with respect to the initial (measured) state. Then, the first control input of the predicted optimal control sequence is applied to the plant. MPC in power electronic and drive systems is distinguished whether a controller takes the discrete nature of power electronic converter (the switches are on or off ) into account or not [2]. Direct or Finite Control Set (FCS) MPC takes the discrete switching states into account. This concept is comparable with direct torque control since both concepts require a relatively high sampling frequencies and have a variable switching frequency. However, FCS-MPC can obtain either a higher current quality or a lower sampling frequency [5]–[8]. Modulator based or Continuous Control Set (CCS) MPC uses PWM or SVM schemes to abstract the integer

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nature of the converter. This approach is similar to vector control since both concepts use a modulator and the constant switching frequency. In comparison, CCS-MPC obtains faster dynamics without overshoots and ringing [3], [4]. Once defined the model predictive controller, there are two methods for implementation; explicit or implicit. In implicit MPC the CFTOC is solved in real-time. Solving the CFTOC problem can require significant computation power, which is particularly demanding at the sampling time scales of power electronic systems. For such systems, explicit MPC has been introduced where the CFTOC problem is solved offline. The CFTOC is solved parametrically [18], [19] as a function of the initial state. For CFTOCs with linear dynamics, affine constraints, and linear or quadratic cost functions it has been shown [17], [20] that the optimal control law is piecewise affine on polyhedra; the state-space is divided into polyhedral regions where each region has a combined feedback and feedforward controller. The main disadvantage of explicit MPC is the excessive storage requirement for complex problems. The main advantage are the relaxed requirements on realtime computation. The computation effort is limited to find the active polyhedron of the stored controller and apply the associated control law. Both, CCS-MPC and FCS-MPC can be applied in a wide range of power electronic systems. In this research, we compare both approaches and point out benefits and drawbacks. MPC can be formulated with several control objectives: current, torque, or power control. Since the objective does not influence significantly the confrontation, we choose the Model Predictive Current Control (MPCC) problem for simplicity. We propose an unified approach to design FCS-MPCC and CCSMPCC for general three-phase neutral-point isolated resistiveinductive loads with internal voltage source. These controllers can be used for grid-connected three-phase power converters with active front-end or permanent-magnet-synchronousmachine drive systems. FCS-MPCC and CCS-MPCC are designed and implemented using the the explicit MPC approach. The controllers are synthesized with MATLAB using MPT [18] and YALMIP [19] and exported in c-code for testing on a experimentally-validated [11]–[13] software-in-the-loop platform.

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We present results comparing the performance of FCS-

MPCC and CCS-MPCC. Steady-state1 performance is measured in terms of current quality, i.e. total harmonic distortion (THD), versus switching frequency. The dynamic performance is evaluated by estimating the resulting control bandwidth. We show that both approaches achieve similar performance. FCSMPCC has slight advantages in dynamics due to the smaller sampling times and the resulting smaller delays. However since CCS-MPCC can work with larger sampling-times, it has lower requirements on the control architecture. This makes it more versatile for application on different systems and power levels. II. A NALYSIS In this section, we show the dynamic models of the system. First, we model the load and the power converter. Then, we combine the models for obtaining an overall systems model, which is used for MPC. A. Load A general three-phase, resistive-inductive load with internal voltage source is described by the three-phase equation vph (t) = Riph (t) + Li˙ ph (t) + eph (t),

(1)

where vph ∈ R3 is the terminal phase voltage, iph ∈ R3 the phase current, eph ∈ R3 the internal voltage source per phase. The parameters R ∈ R+ and L ∈ R+ are the resistance and inductance, respectively. As observed in [10], the effect of the resistance is small compared to the other entities. Thus, we model the system with R ≈ 0 throughout this paper. The equations can be transformed into the αβ0 system using the Clarke transformation   2 −1 −1 √ 1 √ (2) Tαβ0 = 3 − 3 . 0 3 1 1 1 In this research, we focus on neutral point isolated loads, which implies that the zero current i0 = 13 (i1 + i2 + i3 ) ≡ 0. Thus, the system can be described neglecting the zero component and using the αβ component only with the transformation   1 2 √ −1 −1 √ . T= (3) 3 − 3 3 0 We transform the voltage equation (1) into the αβ system state-space representation. The state is iαβ = Tiph ; the controlled input is uαβ = Tuph , and the exogenous input is eαβ = Teph . The state-space dynamic equation is

Fig. 1: Inverter and three-phase resistive-inductive load with internal voltage source where T# is the pseudo-inverse of T. Moreover, the system is discretized for implementation on a digital control platform is aimed. The discrete state-space system is Ts Ts iαβ [k + 1] = iαβ [k] + vαβ [k] − eαβ [k], (5) L L where kTs is the sampling instant with k ∈ N+ and the sampling period Ts ∈ R+ . The controlled input vαβ is used to steer the state iαβ considering (5). The exogenous input eαβ is a rotating vector, which is measured or estimated. Thus, eαβ [k] and its rotational speed ω are known. Both, the magnitude of eαβ and the angular velocity ω are slow varying, i.e. approximately constant for the time scales considered in (5) [4], [7]–[13]. Thus, we write eαβ [k + 1] = Seαβ [k], with the rotation matrix  cos ωTs S= sin ωTs

 − sin ωTs . cos ωTs

(6)

(7)

B. Power Converter In this research, we focus on the two level voltage source inverter (VSI), which is shown in Fig. 1. Each phase is connected to both sides of the DC-link with a semiconductor switch (e.g. IGBT, MOSFET) and an anti-parallel diode. The two switches connected to the same phase are called leg of the converter. Each of the resulting six switches can whether be on or off leading to a large number of possible combinations. Some of them make no sense (the resulting AC voltage is not unique) or are dangerous (short circuit of legs). Neglecting those states, each leg has two combinations: whether the upper and lower switch is closed (leg position 1 and 0, respectively). The second switch of the leg has always the opposite position of the first one2 . Thus, the possible switching states sph of the converter are 3

sph ∈ S = {0, 1} ,

(8)

(4b)

According to this notation, the following equation can be derived from Fig. 1 vc vph = (2sph − 1) , (9) 2

refer to steady-state as operation with constant references and disturbances ignoring the (non steady-state) switching transients.

2 During the transition from position 1 and 0 and vice versa, both switches of a leg are open for the so called interlock time. However, the interlock time is small compared to the sampling time and is neglected.

1 1 i˙ αβ (t) = T T# vαβ (t) − T T# eαβ (t) L L 1 1 = vαβ (t) − eαβ (t), L L 1 We

(4a)

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C. System In this section, we show the overall plant model which is used for synthesizing the MPC current controller. We combine the dynamic equations of the power converter and the load and obtain Ts i[k + 1] = i[k] + Bu[k] + e[k] (13) L

Fig. 2: Power electronic converter, load, and control system with MPC and observer (Obs.) block. The PWM module is used in case of CCS-MPC and it is omitted for FCS-MPC. The external controller sets a current reference r based on the generic reference value ref and generic measurements m.

where vc is the dc-link voltage, which is constant. Neglecting the zero components, the equation becomes vph = sph vc and the corresponding αβ and 0 voltage is vαβ = Tvph = vc Tsph ,   1 1 ksph k1 − , v0 = vc 3 2

(10a) (10b)

where k.k1 is the 1-norm and ksph k1 = 1| sph since sph ≥ 0. However, v0 cannot drive currents. Thus, it is neglected for control design. Introducing a modulator, an average state is applied over the sampling period instead of a discrete state. Thus, the switched function sph is replaced with the duty cycle 3

dph ∈ D = [0, 1] ,

(11)

vαβ = Tvph = vc Tdph ,   1 1 v0 = vc kdph k1 − , 3 2

(12a)

which leads to

(12b)

where kdph k1 = 1| dph since dph ≥ 0. Mathematically, the formulation (12) is the continuous relaxation of the formulation (10). The FCS-MPCC approach uses the discrete (binary) inputs with formulation (10). An appropriate input s is chosen based on the state measurement at the beginning of a sampling period and the binary input is kept for the entire sampling period. Thus, FCS-MPCC manages directly the switching behavior of the power converter. In the CCS-MPCC approach, the continuous (averaged) inputs with formulation (12) are used. The state (current) is measured with synchronous sampling [14], where the average state over the sampling period is directly measured. The input d is chosen based on the average state measurement and the duty cycle is applied through a modulator as combination of switching states such that their average over the sampling period is d. Thus, CCS-MPCC does not manage the switching of the power converter, which is left to the modulator scheme.

where B = vc TLs T. The controlled plant input u (further called simply input) generalizes the controlled input and can be either u = s for FCS-MPCC or u = d for CCS-MPCC. For the sake of compactness, we omit the suffixes αβ and ph The goal of current control is the tracking of a rotating reference current vector r. The reference value r[k] is set by external control loop (DC-link voltage control, speed, power control, etc.) as shown in Fig. 2. The external and internal (MPCC) control loop are decoupled by dynamics. In other words, the system is designed such that the magnitude of r varies slowly compared to the dynamic (13) [4], [7]–[13]. Thus, r can be modeled similar to e, i.e. r[k + 1] = Sr[k]. Since MPC current control aims the reduction of the current control error, we write the system dynamic in terms of the current error x[k] = i[k] − r[k] instead of the current i[k]. Thus, the system dynamics becomes Ts e[k] − (r[k + 1] − r[k]). (14) L Moreover, we introduce the general notation for (known) disturbances w[k] = TLs e[k] − (S − I)r[k] and obtain x[k + 1] = x[k] + Bu[k] +

x[k + 1] = x[k] + Bu[k] + w[k].

(15)

The state i, i.e. x, the input u, and the disturbance w are subject to limitations. In steady-state conditions, the absolute current kik2 (magnitude of the current vector i) has to be below the rated current Ir , i.e. i ∈ I = {i ∈ R2 | kik2 ≤ Ir }. Thus, the reference currents are externally limited to r ∈ I and the condition i ∈ I is satisfied at the equilibrium x = 0. The resulting current error is in the set x ∈ X = I ⊕ (−I) = {x ∈ R2 | kxk2 ≤ 2Ir },

(16)

where ⊕ denotes the Minkowski sum [21]. We note that x ∈ X does not imply i ∈ I. However, we accept short term (transient) violations of the steady-state current constraint. The input constraint u ∈ U is the generalization for u = s ∈ U = S for FCS-MPCC or u = d ∈ U = D for CCS-MPCC. III. S YNTHESIS In this section, we synthesize the MPC current controller. First, the controller is outlined. We define the constrained finite time optimal control (CFTOC) [17] problem and show the MPC algorithm [17]. Then, the controller is implemented using the explicit MPC technique. The CFTOC problem is solved offline and exported as c-code using MPT toolbox [18] with YALMIP [19]. Moreover, we discuss how to obtain the necessary measurements and design choices of the MPC algorithm.

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A. Outline In MPC, the CFTOC problem [17] is solved for finding the feedback control input3 u[k] = u0 .

For being able to implemented the MPCC with the MPT toolbox [18] and YALMIP [19], the batch dynamics [17] xj+1 = x0 +

Problem 1 (CFTOC). minimize U,X

N X j=0

x|j Qxj +

N −1 X

u ¯|j R¯ uj

(17a)

j=0

(17b) (17c)

The CFTOC problem is parametrized by the initial current error x0 = i[k] − r[k] ∈ X , the previous control input u−1 = u[k − 1] ∈ U, and the disturbance sequence W = [w0 , ..., wj , ..., wN −1 ]| with wj ∈ W. The disturbance sequence is computed before solving Problem 1. The CFTOC problem is solved by the input sequence U = [u0 , ..., uN −1 ]| , which produces the predicted states X = [x1 , ..., xN ]| . If the sequence is such that all system constraints are satisfied for j = {0, 1, ..., N }, then the sequence is called feasible. The feasible sequence, which produces the smallest cost, is called optimal and it is denoted with U ? = [u?0 , ..., u?N −1 ]| . According to the MPC policy (or Receding Horizon Control (RHC) policy), the first input of the optimal control sequence is implemented as feedback control law: u(k) = u?0 . Thus, we define the MPC control algorithm as follows. Algorithm 1 (MPC). do at each sampling instant kTs , k ∈ N0 • • • •

Bul +

l=0

subject to xj+1 = xj + Buj + wj ∈ X ; u ¯j = uj − uj−1 ; uj , uj−1 ∈ U;

•

j X

measure and compute the parametrization of Problem 1 obtain U ? by solving Problem 1 if U ? = ∅ then “problem is infeasible” stop apply u[k] = u?0 of U ? to the system wait for the next sampling instant (k + 1)Ts

enddo B. Implementation The controller is implemented using the explicit MPC technique [3], [4]. Problem 1 is solved offline using MPT toolbox [18] with YALMIP [19] for all admissible initial parameterizations. For quadratic cost functions with affine constraints, the feedback control law is piecewise linear. Moreover, the piecewise linear regions depend on the active constraints and can be identified with the initial parametrization. A search-tree is computed for finding the active region and it is exported in c-code together with the piecewise linear feedback laws. Thus, the online computation effort is reduced to find the active region in the search tree and apply the control law which is associated with that region. 3 The suffix . refers to the prediction step j with respect to the actual j sampling instant k, i.e. the step k + j. If the model is exact, we have correspondence between the predicted and measured states, i.e. xj = x[k +j] if u[k + l] = ul for l = {0, 1, ..., j − 1}.

j X

wl

(18)

l=0

is introduced. The advantage of the batch dynamics is that the states are written as function of the parametrization and of the optimization variables of the problem. Let the parametrization vector be p = [x0 , u−1 , w0 , ...wN −1 ]| and the optimization vector be z = [u0 , ..., uN −1 ]| . Then, we can write X = Zx z + Px p with the matrices  B 0 B B Zx =   B B

  ... 0 I 0 I 0 ... 0   ; Px =    ... ... B I 0

(19)

 I 0 ... 0 I I ... 0 .  ... I I ... I

¯ = [¯ Similarly, we define U u0 , ..., u ¯N −1 ]| and ¯ = Zu z + Pu p U with the matrices    0 I 0 ... 0 0 0 −I I ... 0 0  ; Pu =  Zu =     ... 0 0 0 ... −I I PN | The cost function Γ = j=0 xj Qxj + Problem 1 can now be written as

(20)

−I 0

 0 ... 0 0 ... 0 .  ... 0 ... 0

0 PN −1 j=0

u ¯|j R¯ uj of

¯ +U ¯ |R ¯, ¯U Γ = X | QX

(21)

omitting the constant offset x|0 Qx0 , which does not influence the optimization. The matrices are     R 0 ... 0 Q 0 ... 0     ¯ =  0 Q ... 0  ; R ¯ =  0 R ... 0  . Q     ... ... 0 0 ... Q 0 0 ... R Substituting the batch dynamics and the above defined vectors, the cost function can be written as Γ = z | Fz + p| Gz

(22)

The resulting matrices are ¯ x + Z| RZ ¯ x + 2P| RZ ¯ u ; G = 2P|x QZ ¯ u. F = Z|x QZ u u ¯ x + P| RP ¯ u )p is omitted. and the offset p| (P|x QP u Let us now define the feasible trajectories in z and p. Since the MTP toolbox requires affine constraints in the optimization variable z, the state constraint xj ∈ X for j = {1, ..., N } is approximated with a polyhedral set X ≈ {x | Hx ≤ K} [3], [4]. Equivalently, we write

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X ∈ X N ≈ {X | Hx X ≤ Kx }

(23)

TABLE I: Plant Data

where  H 0 Hx =   0

0 ... H ... ... 0 ...

Converter

  0 K K  0  ; Kx =   ;   ...  K H 

Type DC-link voltage Udc Rated current Ir Interlock time Ti

Electrical Machine Type PMSM Rated machine speed ωr 1200rpm Rated torque Tr 6.6N m Inductance (d axis) Ld 20mH Inductance (q axis) Lq 20mH Stator resistance R 636mΩ PM rotor flux ψ 88mW b Pole pairs p 5 Friction constant B 1.7 · 10−3 kgm/s2 Inertia constant J 1.0 · 10−3 kgm2 Control Type CCS1 CCS2 FCS Sampling frequency 2kHz 4kHz 10kHz Switching frequency 1kHz 2kHz variable Prediction horizon 1 1 1 Complexity (number of regions) 33 33 8 Weighting matrix Q I Weighting matrix R 10−3 I

or by substituting the definition of X z ∈ {z | Hx Zx z ≤ Kx − Px p}.

(24)

Moreover, we require uj ∈ U for j = {0, ..., N − 1} or equivalently z ∈ U N . We obtain the polyhedral set U N = DN = [0, 1]3N for CCS-MPCC. For FCS-MPCC, the set is the integer set U N = S N = {0, 1}3N . Therefore, Problem 1 is written in the standard solver form minimize z | Fz + p| Gz

(25a)

z

subject to xj+1 = xj + Buj + wj ∈ X ; Hx Zx z ≤ Kx − Px p; z ∈ U

(25b) N

(25c)

This problem is a static optimization problem in the optimization vector z. It is independent of time and depends on the initial parametrization vector p. This problem can be solved with the MTP toolbox and YALMIP. For CCSMPCC all constraints are affine, which results in a quadratic programming (qp) problem [17]. On the other hand, FCSMPCC leads to a mixed integer quadratic programming (miqp) problem [17] due to the integer set S N . C. Discussion First we discuss how to obtain the required measurements in Algorithm 1. If k is the sampling instant where Algorithm 1 actuates u[k] = u?0 , the algorithm cannot rely on measurements of the same period due to the finite execution time of the digital controller. The resulting unit delay between the measurement (from time instant (k − 1)Ts ) and supposed measurement and actuation (in time instant kTs ) is critical and needs to be compensated [9]. In addition, filtering is required since MPC is sensitive to measurement noise and not modeled high frequency dynamics (e.g. due to parasitic capacitances). Moreover, the internal voltage source is not always accessible for measurement or is preferred to be estimated [22]. For dealing with these issues, a unified approach has been introduced in [11]–[13]. An observer of Luenberger type [23] estimates the state i[k] and disturbance w[k − 1] using the error between measurement i[k −1] and past estimate. Additional integration of the estimation error removes steady-state offsets. Now, we discuss the choice of the design parameters: the prediction horizon N and the matrices Q  0 and R  0. The complexity of solving the CFTOC Problem 1 increases exponentially with the prediction horizon. For explicit MPC, long prediction horizons result in excessive storage requirements. Thus, the prediction horizons N is limited to a few sampling time steps in general [3], [4]. The main goal of MPCC is to minimize x. Thus, we choose Q and R such that u ¯|j R¯ uj is small compared to x|j Qxj . R is mainly introduced to define what happens with the additional

2-level VSI 300V 10A 1µs

degree of freedom of uj compared to xj by penalizing variations of the input. In CCS-MPCC, the additional degree of freedom may cause trouble to the solver and lead to undesired results in the sense that the actuation of d is not necessarily continuous. Introducing R  0 the actuation of d is smooth. In FCS-MPCC, R penalizes the variation of the switching transitions of the legs and can be used to reduce the switching frequency. Moreover, the passive inputs s = [0, 0, 0]| and s = [1, 1, 1]| have the same effect on the states. With R = 0, they are selected whether randomly or one of them is never selected. Thus, R  0 avoids an artificial augmentation of the switching frequency and the passive input is selected, which can be reached with the lower number of switching transitions. IV. R ESULTS In this section the FCS-MPCC and CCS-MPCC approaches are compared in terms of performance. The MPC controllers have been implemented for the permanent-magnetsynchronous-machine drive system using MPT and YALMIM. The controllers are exported in c-code and they are applied to a experimentally validated [11]–[13] software-in-the-loop platform platform for evaluation. The characteristics of the drive system are shown in Table I. The controllers are designed to obtain approximately the same switching frequency, i.e. to produce approximately the same switching losses. The FCS-MPCC type controller has been implemented with a sampling frequency of 10kHz. Since the FCS controllers do not achieve a constant switching frequency, it is measured and it is shown to be in the order of 1kHz, i.e. 2kHz. For comparison, two CCS controllers have been implemented with 2kHz and 4kHz sampling frequency and obtain a switching frequency of 1kHz and 2kHz, respectively. The characteristics of the controllers are shown in Table I.

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A. Steady-State In this section, the steady-state behavior is evaluated. In this condition, a high current quality using a possibly low switching frequency is aimed. Current quality is measured in terms of total harmonic distortion (THD) with respect to the rated current qP 50 h=2 ih THDph = (26) Ir where ih is the amplitude of the harmonic of order j. The THD is calculated as arithmetic mean of the THD per phase THDph . Moreover, we show the normalized current ripple

and the switching frequency, which is obtained in different operation points. The current quality has been evaluated at the the mechanical speeds n = {0, 300, 600, 900, 1200, 1500}rpm with 5N m (75%) load torque, which corresponds approximately to the modulation indexes m = {0, 0.25, 0.50, 0.75, 1.00, 1.25}, which is defined as m = 2k¯ uk2 − 1

(27)

Fig. 3: CCS1 (fsw = 1kHz) steady-state current waveform and spectrum dependent on modulation index, from top: m = 0.25, m = 0.75, m = 1.25

where u ¯ is the average control action (duty cycle or voltage vector). Examples of the current waveforms and the corresponding spectra are shown in Fig. 3, Fig. 4, and Fig. 5. The CCSMPCC spectra in Fig. 3 and Fig. 4 show emphasized switching (PWM) harmonics and low noise. Contrarily, the FCS-MPCC spectra in Fig. 5 contains above all noise and integer-multiples of of the fundamental frequency are emphasized in overmodulation. In Fig. 6, we compare the CCS-MPCC and FCS-MPCC switching frequency and the current quality, i.e. THD and normalized current ripple. The results are computed in steadystate using a 10s time window. The CCS-MPCC concept has a constant switching frequency, which decreases slightly in over-modulation (passive vectors are not always applied). The current ripple of CCS-MPCC is low for low modulation indexes but increases rapidly with the modulation index. On the other hand, the FCS-MPCC switching frequency varies with the modulation index and has a peak at approximately m = 0.75. However, an approximately constant current quality (for m > 0) is obtained. It is noted that the steady-state performance of FCS-MPCC can be increased by planning the current trajectory. This involves further optimization, which adds significant complexity, and requires a good knowledge of the plant model [7], [9].

Fig. 4: CCS2 (fsw = 2kHz) steady-state current waveform and spectrum dependent on modulation index, from top: m = 0.25, m = 0.75, m = 1.25

Fig. 5: FCS steady-state current waveform and spectrum dependent on modulation index, from top: m = 0.25, m = 0.75, m = 1.25

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period and both, delays and estimated bandwidth, must be taken into account for describing dynamically the controller. In conclusion, both CCS-MPCC and FCS-MPCC achieve fast dynamics and differ only by the actuation delays. CCSMPCC achieves significantly faster dynamics than linear vector control without overshoots and ringing [4]. This is sufficient for most applications. For even faster dynamics, the FCS-MPCC method can be employed.

Fig. 6: Steady-state switching frequency, THD, and current ripple

In conclusion, both concepts achieve similar steady-state performance. Whether FCS-MPCC or CCS-MPCC is advantageous depends on the application, i.e. if approximately constant current quality or switching frequency is preferable. Also the current spectrum will influence which MPCC is preferable since the choice determines the design of harmonic filters and the noise emission. Fig. 7: CCS1 (fsw = 1kHz) reference step (1A) without control saturation

B. Dynamic In this section, the dynamic behavior of the controllers is shown. For characterizing the controllers, a reference-step is applied and the closed loop response is evaluated. For large reference-steps, the control action u saturates and the dynamics (rise time) is imposed by the control constraints. Thus, small-signal reference-steps are actuated, which corresponds to a reference-step of 1A for CCS-MPCC. For FCS-MPCC, the integer nature of the control input can be interpreted as limitations which are always active. Anyway, the 1A step is applied also to this controller for comparison. The results are shown in Fig. 7, Fig. 8, and Fig. 9. The bandwidth bw of the system is estimated using the 10% to 90% rise time tr with the equation bw =

0.34 . tr

(28)

As Fig. 7 to Fig. 9 suggest, the three controllers have approximately the same estimated bandwidth bw = 4kHz. However, this result does not take the actuation delays into account. A unit delay is introduced from the instant that the reference value is set until the control actuation is applied to the plant4 . Moreover, the PWM module (in case of CCS-MPCC) does not apply instantaneously the control input but introduces a delay, which can be approximated with half a sampling period. This delay can be observed in Fig. 7 and Fig. 8. In case of the CCS-MPCC, the rise time is small compared to the delays and the control dynamics can approximated with the delays in the system. For the FCS controller, delays are smaller due to the smaller sampling

Fig. 8: CCS2 (fsw = 2kHz) reference step (1A) without control saturation

4 Delays are independent of MPC and are introduced by digital controllers due to the finite execution time of the control algorithm.

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Fig. 9: FCS reference step (1A)

V. C ONCLUSION In this research, continuous control set (CCS) and finite control set (FCS) model predictive current control (MPCC) for power electronic and drive systems have been compared in terms of steady-state, i.e. current-quality versus switching frequency, and dynamic performance. A unified formulation is proposed to design explicit CCS-MPCC and FCS-MPCC. The controllers are developed and implemented using MATLAB with MPT and YALMIP. Both controllers are exported in ccode for evaluation on a experimentally validated software-inthe-loop platform platform. In steady-state, the total harmonic distortion (THD), the normalized current ripple, and the switching frequency is evaluated for different operation points. Moreover, the current waveforms and spectra are shown for both concepts. CCSMPCC achieves a spectrum with PWM switching harmonics and low noise. The switching frequency is constant but the THD and the current ripple increase nearly linearly with the modulation index. The FCS harmonic spectrum is nearly white noise and the THD and current ripple is approximately constant. However, the switching frequency depends on the operation point. The MPCC dynamics is evaluated by estimating the bandwidth through the rise time, where both concepts achieve the same results. However, when taking system delays into account, FCS-MPCC has advantages due to the smaller sampling times. Both concepts achieve similar steady-state and dynamic performance. Whether FCS-MPCC or CCS-MPCC is advantageous depends on the application, i.e. if approximately constant current quality or switching frequency is preferable. Also the current spectrum will influence which MPCC is advantageous since the choice determines the design of harmonic filters and the noise emission. The dynamic performance of both concepts is sufficiently high for most applications. However, since CCS-MPCC can work with larger sampling-times, it has lower requirements on the control architecture. This makes it more versatile for application on different systems and power levels.

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