Finite Set Model Predictive Control with On-line

Electrical Engineering manuscript No. DOI: 10.1007/s00202-017-0606-3

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Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters Mohamed Abdelrahem1,2 · Christoph Michael Hackl3 · Ralph Kennel1

Received: / Accepted: Springer-Verlag Berlin Heidelberg

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Abstract This paper proposes a finite set model predictive control (FS-MPC) system for active front-end (AFE) converters in combination with an extended Kalman filter (EKF) for on-line parameter estimation to overcome the issues of parameter uncertainty and measurement noise. The FS-MPC performance of such converters is largely affected by variations in the model impedance (filter impedance and grid impedance), especially for systems with low short circuit ratios (SCR). Therefore, an EKF is used to estimate these model parameters on-line. Moreover, the EKF is used to filter out measurement noise in the feedback variables. For implementation of the FS-MPC, the discrete-time models of the AFE converter and the filter are derived using two discretization methods (forward Euler method and Taylor series expansion). The observability matrix of the linearized model is computed and the observability is checked for each time instant to verify that the EKF is working properly. Moreover, the delay time due to the digital calculation is compensated. The performance of the proposed method is illustrated via simulation results. Keywords Finite set model predictive control · active front-end converters · extended Kalman filter · noise rejection · state estimation 1

Institute for Electrical Drive Systems and Power Electronics, Technical University of Munich (TUM), Germany. E-mail: [email protected], [email protected] 2 Electrical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt. 3 Munich School of Engineering, Research Group “Control of Renewable Energy Systems (CRES)”, TUM, Munich, Germany. E-mail: [email protected]

Notation

N, R, C are the sets of natural, real and complex numbers. x ∈ R or x ∈ C is a real or complex scalar. x ∈ Rn (bold) is a real valued vector with n ∈ N. x> is the transpose and √ kxk = x> x is the Euclidean norm of x. 0n = (0, . . . , 0)> is the n-th dimensional zero vector. X ∈ Rn×m (capital bold) is a real valued matrix with n ∈ N rows and m ∈ N columns. O ∈ Rn×m is the zero matrix. diag(a1 , . . . , an ) ∈ Rn×n is 2 3 a diagonal matrix with entries a1 , . . . , an . xy z ∈ R or ∈ R is a space vector expressed in either phase abc-, stationary fixed s-, or arbitrarily rotating k-coordinate system, i.e. y ∈ {abc, s, k}, and may represent voltage u, flux linkage ψ or current i, i.e. x ∈ {u, ψ, i}. E{x} or E{X} is the expectation value of x or X, respectively.

1 Introduction

AFE converters are widely used in industrial applications, such as advanced machine drives with regenerative breaking capability or renewable energy systems [1], [2]. AFE converters are characterized by the possibility of a bidirectional power flow, a flexible DClink voltage regulation, and sinusoidal input/output currents with low harmonic distortion [2]. Moreover, the amount of reactive power drawn from the source can be controlled in order to ensure a unity power factor, or to compensate for a lack of reactive power in the grid [2]. The most popular approach for controlling the AFE converters is voltage-oriented control (VOC), where a dual-loop control design is often preferred. An inner current control loop is used to control the currents injected into the grid, and an outer voltage loop is used

2

M. Abdelrahem, C. Hack, and R. Kennel

Load

sa

udc

sb

sc

i abc Rf L f f

Rg L g u a o

uob uoc

C dc

sa

sb

sc

filter

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to regulate the DC link voltage [3]. A different approach to VOC is direct power control (DPC), which has been successfully applied to power converters. The main goal of DPC is the direct regulation of the active and reactive power without the use of a pulse width modulator [3]. However, with the development of faster and more powerful micro-processors, the implementation of new and more complex control schemes is possible. Some of these new control schemes for power converters include fuzzy logic, sliding mode control, predictive control, neural networks, and other advanced control techniques [4].

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Predictive control comes with several advantages [4]-[9]: Concepts are considered intuitive and easy to understand. Predictive controllers can be applied to a variety of systems, constraints and nonlinearities can be included easily, the multi-variable case can be considered, and the resulting controllers are simple to implement (however, often computationally demanding). One of the most promising predictive controllers for power converters is the so-called model predictive control (MPC) [4]-[9], in which a model of the system is considered in order to predict the future behavior of the variables. The MPC techniques have been classified into two main categories [9]: continuous set MPC (CSMPC) and finite set MPC (FS-MPC). In the first group, a modulator generates the switching states according to the continuous output of the model predictive controller. On the other hand, the FS-MPC approach takes advantage of the limited number of switching states of the power converter to solve the optimization problem. A discrete model is used to predict the behavior of the system for all admissible switching states over the prediction horizon. The switching action that minimizes a predefined cost function is finally selected and applied at the next sampling instant. The main advantage of FS-MPC lies in the direct application of the control action to the converter without the need of a modulation stage [4]-[21]. However, the performance of the FSMPC is dependent on accurate prediction of the future behavior of the system. Inaccurate parameters used for the controller can lead to an inaccurate prediction of the future behavior due to an incorrect system model, which might results in the selection of incorrect switching states (e.g. deteriorating the control performance or even endangering stability). Therefore, it is essential to construct an exact system model with accurate model parameters to avoid performance degradation and instability of the control system [22]. FS-MPC for direct power control (DPC) or current control for AFE converters was introduced in the literature in [15]-[21]. In these papers, [15]-[21], the model parameters are considered to be constant.

Grid

DC Link

Fig. 1: Active front-end converter topology.

Several methods have been proposed in the literature for the estimation of the filter or the grid inductance (impedance) [23]-[29]. Reference [23] proposed a virtual flux-based predictive DPC with on-line inductance estimation for AC/DC converters. An analytic method for the estimation of the coupling inductance of AFE converters is proposed in [24]. The authors in [25] proposed a fuzzy logic controller based method to compensate for the unknown coupling inductance used in an AFE converter control strategy. However, in these papers [23][25], the coupling resistance is not considered. A grid impedance estimation method based on variations of active and reactive power was proposed in [26]. In [27], an extended Kalman filter (EKF) is used to estimate the inductive and resistive parts of the grid impedance. A grid parameter estimation method based on the assumption that the grid voltage magnitude is equal at two consecutive sampling instants was proposed in [28]. The main drawback of these papers [26]-[28] is that the filter impedance is considered to be constant. A predictive DPC with an adaptive on-line parameter identification technique for AFE converters was proposed in [29]. The proposed parameter identification technique is based on a least squares method to estimate the filter inductance and resistance for the grid-connected AFE converter; but grid inductance and resistance were not considered in [29]. In this paper, a FS-MPC with on-line parameter estimation for AFE converters is proposed to overcome the parameter uncertainty problem. The proposed estimation algorithm is based on the use of an extended Kalman filter (EKF) to estimate the system parameters (filter and grid impedance) on-line and reject the noise in the current measurements. The discrete-time model of the AFE converter including filter and grid is derived using two different discretization methods: the forward Euler method and the Taylor series expansion. The observability matrix of the linearized model is computed and the observability is checked on-line for each sampling instant. Moreover, the induced delay due to the digital calculation is compensated. The performance of

Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters

Table 1: Switching states sabc and AFE converter output voltage ufs . β > ufs = (uα f , uf ) s uf (0, 0, 0) = (0, 0)> ufs (1, 0, 0) = ( 32 udc , 0)>

sa 0 1

sb 0 0

sc 0 0

1

1

0

ufs (1, 1, 0) = ( 31 udc ,

ufs (0, 1, 0) ufs (0, 1, 1) ufs (0, 0, 1) ufs (1, 0, 1) ufs (1, 1, 1)

√

3 u )> 3√ dc 1 (− 3 udc , 33 udc )> (− 32 udc , 0)> √ (− 13 udc , − 33 udc )> √ ( 31 udc , − 33 udc )> >

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the proposed method is illustrated via simulation results. This paper is organized as follows: Section 2 presents the mathematical model of the AFE converter and Sec. 3 covers the discretization methods and explains the proposed FS-MPC. The proposed EKF and observability of the system are explained in Sec. 4, while the simulation results are given in Sec. 5. Finally, the paper is concluded in Sec. 6.

3

2 Modeling of the AFE converter

0 0

1 1

0 1

0

0

1

1 1

0 1

1 1

= =

=

= = (0, 0)

In Fig. 1, a grid-connected three phase AFE converter is shown. It is connected to the grid through a RL-filter with resistance Rf [Ω] and inductance Lf [Vs/A]. The grid is represented as an ideal voltage source with threephase voltage vector uoabc = (uoa , uob , uoc )> [V], resistance Rg [Ω] and inductance Lg [Vs/A]. The three-phase current vector ifabc = (ifa , ifb , ifc )> [A] flows from the grid to the AFE converter. The load is connected in parallel to the DC-link capacitance Cdc [As/V] with DC-link voltage udc [V]. Invoking Kirchhoff’s voltage law at the AC side of the AFE converter leads to the following model [30]:  d abc uoabc (t) = (Rf + Rg )ifabc (t) + (Lf + Lg ) dt if (t)   +ufabc (t), ifabc (0) = 03 .  {z } | 

Considering all the possible combinations of the switching state vector sabc : eight switching states, and consequently, eight voltage vectors are obtained. Note that, due to the two zero switching vectors (0, 0, 0) and (1, 1, 1), only seven different voltage vectors are obtained, see Table 1. Equation (2) can be expressed in the stationary reference frame as xs = (xα , xβ )> = TC xabc by using the Clarke and Park transformation (see, e.g., [30]), respectively, given by (neglecting the zero sequence) " #   1 1 2 1− cos(φ) sin(φ) s s abc k 2 −√2 √ x = x x &x = − sin(φ) cos(φ) 3 0 23 − 23 {z } | | {z }

(1)

The voltage equation (2) can be rewritten in the αβ reference frame as follows d uos (t) = Rt ifs (t) + Lt ifs (t) + ufs (t), ifs (0) = 02 . (6) dt The voltage orientation control is achieved by aligning the d-axis of the synchronous (rotating) reference frame with the grid voltage vector uos which rotates with the grid angular frequency ωo (under ideal conditions, i.e. constant grid frequency fo > 0, it holds that ωo = 2πfo rad s is constant). Applying the (inverse) Park transformation with TP (φo )−1 as in (5) with Z t φo (t) = ωo (τ )dτ + φ0o , φ0o ∈ R

=:TP (φ)−1

=:TC

initial values

The values (Rf +Rg ) and (Lf +Lg ) represent the overall system resistance and inductance and can be expressed by the total resistance and total inductance, i.e., Rt := Rf + Rg and Lt := Lf + Lg , respectively. Therefore, equation (1) can be rewritten as: d abc i (t)+ufabc (t), dt f

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uoabc (t) = Rt ifabc (t)+Lt

ifabc (0) = 03 | {z } initial values

(2)

Here ufabc = (ufa , ufb , ufc )> [V] is the output voltage of the AFE converter. This voltage can be expressed as: ufabc (t) =

1 udc T abc sabc 3

(3)

where sabc = (sa , sb , sc )> ∈ {0, 1}3 is the switching state vector of the AFE converter and T abc is the transformation matrix describing the relation between switching state vector and output phase voltage vector of the converter and is given by (for a balanced system):   2 −1 −1 T abc = −1 2 −1 (4) −1 −1 2

(5)

e

to the voltage equations (6) yields the description in the rotating reference frame    0 −1 d k uok (t) = Rt ifk (t) + Lt dt if (t) + ωo Lt ifk (t)  1 0  +ufk (t), ifk (0) = 02 , (7) where uok = (ˆ uo , 0)> with grid voltage amplitude u ˆo := s kuo k.

4

M. Abdelrahem, C. Hack, and R. Kennel

Model of physical system (hardware emulation) grid

Load

Rt Lt

u dc , ref i

q f ,ref

udc

DC Link

sa sb s c

PI

[k ]

u oabc

i abc f

al Co py

udc

C dc

uo[k]

o

idf ,ref [k  2]

idf ,ref [k ]

Lag. q Ext. i f ,ref [k  2]

i df [k  2]

uod [k ]

/ dq u q [k ] o

ud [k 1]

Lag. oq u [k 1] Ext. o

abc / 

Cost Function sabc iqf [k  2]

Prediction Model

udc

o

i  f

u o

udc Extend Kalman Filter Rˆt Lˆ t iˆf [k 1]  / dq

iˆfdq[k 1]

o

Control system (digital implementation)

Fig. 2: Block diagram of the proposed FS-MPC with on-line parameter estimation for AFE converters. 3 Discretization & Finite Set Model Predictive Control 3.1 Discretization Methods

The FS-MPC approach uses a discrete-time model for predicting the filter current at a future sample period. Various techniques for discretization of nonlinear systems exist [31]-[33]. Usually, it is necessary to make a trade-off between modeling accuracy and complexity. In this paper, two discretization methods will be used. 3.1.1 Euler Method:

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The simplest method to discretize a nonlinear system is the Euler (Forward) Approximation [31]-[33]. It is by far the most often utilized discretization approach in the literature [4]-[21] (concerning electrical drives and power electronics). For sufficiently small sampling time Ts  1 (compared to the smallest time constant of the considered system), the following holds d x[k] := x(kTs ) ≈ x(t) and dt x(t) = x[k+1]−x[k] for Ts all t ∈ [kTs , [k + 1]Ts ) and k ∈ N ∪ {0}. Hence, applying the forward Euler method to the time-continuous model (7), leads to the discrete-time model (difference equation) of the AFE converter: s Rt ifd [k + 1] = (1 − TL )ifd [k] + ωo Ts ifq [k] t Ts (uod [k] − ufd [k]) +L t q s Rt if [k + 1] = (1 − TL )ifq [k] − ωo Ts ifd [k] t Ts + Lt (uoq [k] − ufq [k])

        

3.1.2 Series Expansion:

The idea of using Taylor series expansion for the discretization of nonlinear systems was already successfully used in [31]-[33]. According to the Taylor series expansion x[k + 1] can be written as x[k + 1] = ∞ i i P Ts d x(t) x[k] + i! dti . Setting i = 2 and assuming that the i=1

input uok (t) is constant over one sampling period Ts , the discrete-time model of the AFE converter based on the Taylor series expansion can be written as follows:  Ts2 Rt2 ωo2 Ts2 d s Rt  + ifd [k + 1] = (1 − TL )i [k] −  2 f  2 2Lt t  2  ωo Ts Rt q   +(ωo Ts − Lt )if [k]    2  Ts Rt Ts d d  +( Lt − 2L2 )(uo [k] − uf [k])   t  2 2  ωo Ts q  + 2Lt (uoq [k] − uf [k]) (9) 2 2 2 2 Ts R t ωo Ts q s Rt  ifq [k + 1] = (1 − TL + )i [k]  2 − f 2  2L t t   ω T 2R   −(ωo Ts − o Lst t )ifd [k]    2  T R q Ts t q s  +( Lt − 2L2 )(uo [k] − uf [k])   t  2 2  ωo Ts  d d − (u [k] − u [k]) 2Lt

o

f

A direct comparison of the complexity of the models (8) and (9) illustrates the simplicity of the forward Euler method (highlighting its wide usage in the literature.) 3.2 Finite Set Model Predictive Control

(8)

Real-time implementation of the predictive control methods usually induces a one-step delay caused by the update mechanisms in modern digital controllers.

Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters

To compensate for the one-step delay, the value at the [k + 2]th instant should be used instead of the [k + 1]th instant [7]. Therefore, (8) and (9) can be rewritten as: + 2] =

ifq [k + 2] =

s Rt (1 − TL )ifd [k + 1] + ωo Ts ifq [k t Ts d + Lt (uo [k + 1] − ufd [k + 1]) s Rt (1 − TL )ifq [k + 1] − ωo Ts ifd [k t Ts (uoq [k + 1] − ufq [k + 1]) +L t

and ifd [k

+ 2] =

ifq [k + 2] =

 + 1]    

ikf,ref [k + 1] = 3ikf,ref [k] − 3ikf,ref [k − 1] + ikf,ref [k − 2]. (14)

+ 1]    

(10)

Using the same idea, the future grid voltage vector uok [k + 1] can be predicted as follows



uok [k + 1] = 3uok [k] − 3uok [k − 1] + uok [k − 2].

(11)

respectively. In this paper, the following simple cost function (with soft constraint) is used + 2] − ifd [k + 2]| + |iqf,ref [k + 2] − ifq [k q if ifd [k + 2]2 + ifq [k + 2]2 ≤ imax

d g = |i + 2]| f,ref [k 0 q + ∞ if ifd [k + 2]2 + ifq [k + 2]2 > imax ,

(12)

Fin

where idf,ref [k+2] and iqf,ref [k+2] are the reference values of the filter currents. The cost function is evaluated over a prediction horizon of two (i.e. prediction horizon over two time steps) to compensate for the computation delay due to the digital implementation of the FS-MPC scheme. The schematic diagram of the proposed FS-MPC techique for AFE converters is illustrated in Fig. 2. The prediction model (10) and (11) are evaluated for each of the possible seven different voltage vectors, giving seven different current predictions. The switching vector sabc which minimizes the cost function (12) will be applied at the next sampling instant. However, the future reference current iks,ref [k + 2] value is unknown. Therefore, it has to be predicted from the present and previous values of the current reference using the nth order formula of the Lagrange extrapolation (as proposed in [7]):   n X k n−l n + l if,ref [k + 1] = (−1) ikf,ref [k + l − n]. (13) l l=0

The reference current ikf,ref [k + 2] can be calculated by shifting (14) forward in time, i.e.: ikf,ref [k +2] = 6ikf,ref [k]−8ikf,ref [k −1]+3ikf,ref [k −2] (15)

Ts2 Rt2 ωo2 Ts2 d  + 2L (1 − 2 −  2 )if [k + 1]  t  2  ω T R   +(ωo Ts − o Lst t )ifq [k + 1]    2  Ts Rt Ts d d +( Lt − 2L2 )(uo [k + 1] − uf [k + 1])   t  2 2  ωo Ts q  q + 2Lt (uo [k + 1] − uf [k + 1]) 2 2 2 2 Ts Rt ωo Ts q s Rt  + 2L (1 − TL 2 − 2 )if [k + 1]   t t  2  ωo Ts Rt d   −(ωo Ts − Lt )if [k + 1]    2  Ts Rt q Ts q  +( Lt − 2L2 )(uo [k + 1] − uf [k + 1])  t   ωo2 Ts2  d d − 2Lt (uo [k + 1] − uf [k + 1]) Ts Rt Lt

For n = 2, using (13), the future reference current ikf,ref [k + 1] can be predicted by

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ifd [k

5

(16)

The value of the d-axis reference current idf,ref [k] is obtained from an outer DC-link control loop as shown in Fig. 2. The reference value of the DC-link voltage udc,ref is compared with the measured value udc and the error is processed by a proportional-integral (PI) controller producing the d-axis reference current idf,ref [k]. The DC-link PI controller is tuned according to the guidelines provided in [30] to ensure stability.

4 Extended Kalman Filter and Observability 4.1 Extended Kalman Filter

Since the late 1960s, the Kalman filter has received significant attention from various fields in industry and academia and played a key role in many engineering disciplines for trajectory planing, state and parameter estimation, signal processing, etc. [34]-[36]. The EKF is a nonlinear extension of the Kalman filter for nonlinear systems and is designed based on a discrete-time nonlinear system model [37]. FS-MPC is sensitive to variation in the model (filter+grid) parameters. Even small variations / uncertainties in the model parameter can result in instability of the closed-loop system or can deteriorate the control performances drastically. Therefore, an accurate knowledge of the model parameters is essential. In this paper, an EKF is used to estimate the filter parameters Rt and Lt , respectively. Both model parameters can change due to heat, aging, and/or saturation. For the design of the EKF, the derivation of a compact (nonlinear) state space model of the AFE converter of the form d x = g(x, u), dt

x(0) = x0 ∈ R4

and

y = h(x), (17)

6

M. Abdelrahem, C. Hack, and R. Kennel

Solving (6) for model (17) with

d α dt if

and

d β dt if

yields the nonlinear

 Rt α α  − Lt if + L1t (uα o − uf )  − Rt iβ + 1 (uβ − uβ )  o f  Lt f Lt g(x, u) =    0 0  1000 h(x) = x. 0100 | {z }

      and     

=:C=[I 2 , O 2×2 ]∈R2×4

(19)

         

For discretization, again the forward Euler method with sampling time Ts [s] is applied to the time-continuous model (17) with (18) and (19). Therefore, the nonlinear discrete-time system model can be written as  =:f (x[k],u[k])   }| { z (20) x[k + 1] =x[k] + Ts g(x[k], u[k]) +w[k], and   y[k] =h(x[k]) + v[k],

Fin

where the random variables w[k] := (w1 [k], . . . , w4 [k])> ∈ R4 and v[k] := (v1 [k], v2 [k])> ∈ R2 are included to model system uncertainties and measurement noise, respectively. Both are assumed to be independent (i.e., E{w[k]v[j]> } = O 4×2 for all k, j ∈ N), to have zero mean (i.e. E{w[k]} = 04 and E{v[k]} = 02 for all k ∈ N) and to have normal probability dis  tributions (i.e., p(γi ) =

Algorithm 1: Extended Kalman filter Step I: Initialization for k = 0 ˆ [0] = E{x0 }, P 0 := P [0] = E{(x0 − x ˆ [0])(x0 − x ˆ [0])> }, x
−1 K 0 := K[0] = P [0]C > CP [0]C > + R Step II: Time update (“a priori prediction”) for k ≥ 1 (a) State prediction ˆ − [k] = f (ˆ x x[k − 1], u[k − 1]) (b) Error covariance matrix prediction P − [k] = A[k]P [k − 1]A[k]> + Q where A[k] = ∂f (x,u) − ∂x

al Co py

is required. Therefore, the state vector x, the output (measurement) vector y and the input vector u are introduced as follows   > β  4 α  x = if , if , Rt , Lt ∈R ,     > β 2 α (18) y = if , if ∈R ,    >   β α β u = uα ∈ R2 .  o − uf , uo − uf

σγi 2

1 √

2π

exp

−(γi −E{γi })2 2 2σα

with σγ2i := E{(γi − E{γi }) } and γi ∈ {wi , vi }). For simplicity, it is assumed that the covariance matrices are constant, i.e., for all k ∈ N the following holds Q := E{w[k]w[k]> } and R := E{v[k]v[k]> } > 0. Note that Q and R must be chosen positive semidefinite and positive definite, respectively. Since system uncertainties and measurement noise are not known a priori, the EKF is implemented as follows
ˆ [k + 1] =f (ˆ ˆ [k] , x x[k], u[k]) + K[k] y[k] − y ˆ [k] =h(ˆ ˆ [k]. y x[k]) = C x

and



(21) where K[k] is the Kalman gain (to be specified in Alˆ and y ˆ are the estimated state and gorithm 1) and x

ˆ [k] x

Step III: Verification of (local) observability k ≥ 1:
no [k] := rank S o [k] with S o [k] as in (22) Step IV: Computation of Kalman gain for k
≥ 1 −1 K[k] = P − [k]C > CP − [k]C > + R Step V: Measurement update (“correction”) for k ≥ 1 (a) Estimation update with measurement ˆ [k] = x ˆ − [k] + K[k](y[k] − h(ˆ x x− [k])) (b) Error covariance matrix update P [k] = P − [k] − K[k]CP − [k] Step VI: Go back to Step II.

output vector, respectively. The recursive estimation algorithm of the EKF implementation is listed in Algorithm 1 [36]. The EKF achieves an optimal state estimation by minimizing the covariance of the estimation error for each time instant k ≥ 1. In [38], general guide lines are given how to select the entries of Q and R. Following these guide lines, the following choices have been made: Q = diag{0.02, 0.02, 8 · 10−4 , 10−7 }, R = diag{1, 1}, P 0 = diag{1, 1, 0.001, 10}, and x0 = (0.01, 0.01, 0.01, 0.001)> .

4.2 Observability

The observability of a linear system can be verified by computing the observability matrix and its rank. For nonlinear systems, it is possible to analyze the observability “locally” by analyzing the linearized model around an operating point [39]. The observability matrix of the linearized model of the considered AFE converter as in (20) is given by   C  CA[k]  8×4  S o [k] :=  (22) CA[k]2  ∈ R , CA[k]3 where A[k] is computed on-line for each sampling instant k ≥ 0 (see Algorithm 1). The pair {A[k], C} (i.e., the linearized model of the AFE converter) is observable if and only if the observability matrix S o [k] has

Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters

iqf ,ref

FCS-MPC (control system)

P[kW ] & Q

P

Q

704

udc AFE abc converter uo (hardware emulation) abc if

Fig. 3: Simulink model of the adopted system.


full rank, i.e., rank S o [k] = 4 for the considered AFE converter as in (20). To check “local” observability, the rank of the observability matrix S o [k] is computed numerically for each sampling instant k ≥ 0 in Step III of Algorithm 1.

5 Simulation Results and Discussion

700

u dc , ref

694

692 25 20 15 10 5.0 0.0 -5 400 200 0

d

q

iaf[ A] & uoa[V ] i f & i f [ A ]

sa sb sc

Value 400[V] 50[Hz] 0.05[Ω] 2[mH] 0.2[Ω] 10[mH] 3[mF] 700[V] 40[µs] 1[µs]

idf ,ref

i df

i qf ,ref

-200 -400

0.0

10

20

30

40

u dc

i qf

10i af

u oa

50

time [ms]

Fig. 4: Simulation results for the proposed FS-MPC scheme (plotted quantities from top to bottom): Output active P and reactive Q power, reference udc,ref and actual udc DC-link voltage, reference idf,ref , iqf,ref and actual idf , iqf currents (in d/q), and phase current iaf and grid phase voltage uao (of phase a).

5

4

%Ih / I 1

udc,ref

Nomenclature uo fo Rg Lg Rf Lf Cdc udc Ts Tsim

10 8 6 4 2 0 -2

al Co py

Name Grid line-to-line voltage Normal frequency Grid resistance Grid inductance Filter resistance Filter inductance DC capacitor DC-link voltage Sampling time Simulation step

udc[V ]

Table 2: AFE converter parameters.

7

3

2

1 0

0

Fin

A simulation model of a 10kW AFE converter is implemented in Matlab/Simulink. The Simulink model is illustrated in Fig. 3, which consists of two parts: (i) the model of the physical system (hardware emulation) and (ii) the control system (digital implementation). The details of each part are illustrated in Fig. 2. The system parameters are listed in Table 2. The implementation of the finite set model predictive control scheme is as in Fig. 2. The simulation results are shown in Figures 4-12. Fig. 4 shows the dynamic performance of the proposed FS-MPC. The proposed MPC scheme achieves accurate and fast reference tracking of the reference values udc,ref , idf,ref , and iqf,ref , respectively. Moreover, the AFE converter operates with unity power factor (see Fig. 4). Fig. 5 shows the harmonic spectrum of the AFE converter current iaf of phase a. The total harmonic distortion (THD) is 4%, which match the IEEE-519 standards for the admissible total harmonic distortion [40] in low voltage grids. Fig. 6 shows the simulation results of the proposed EKF. Despite large initial errors, the parameter estimation is fast; in particular, for the total inductance.

10

20 30 Harmonic order

40

50

Fig. 5: Harmonic spectrum of the AFE phase current iaf (of phase a).

ˆ t and The initial estimation error ein of the resistance R ˆ t is 90% and 75%, respectively. The delay inductance L time td to reach the steady-state value of the resistance ˆ t and inductance L ˆ t is 10ms and 1.2ms, respectively. R The steady state estimation error ess of the resistance ˆ t and inductance L ˆ t is 2.25% and 1.15%, respectively, R which is very small (almost zero), see Fig. 6. Moreover, the observability matrix has full rank for all times,
i.e. rank S o [k] = 4 for all k ≥ 0 as shown in Fig. 6. Fig. 7 shows the control performance of the proposed MPC scheme with EKF during step changes in active and reactive output power P and Q, respectively. According to Fig. 7, the proposed FS-MPC schemes shows very fast closed-loop dynamics. Moreover, it is capable of holding the DC-link voltage almost constant for step changes in active or reactive power. Moreover, idf and iqf follow their references idf,ref and iqf,ref almost

M. Abdelrahem, C. Hack, and R. Kennel

iˆ f

i f

iˆ f

0.2

10

ein

5

696 25 20 15 10 5 0 -5

t

td

ess

Lt Lˆ t

i f & i f [ A]

0 5 4 3 2 1 0 0.0

10

20

0 -5 704 700

30 time [ms]

40

50

u dc , ref

u dc

Fig. 6: Simulation results for the proposed estimation algorithm with EKF (plotted quantities from top to β ˆα ˆβ bottom): Actual iα f , if and estimated if , if currents (in ˆ t resistance, total Lt and α/β), total Rt and estimated R ˆ estimated Lt inductance, and rank of the observability matrix (rank(S o [k]) = 4 holds for k ≥ 0).

Fin

instantaneously, respectively. Furthermore, the EKF assures an accurate estimation of the total resistance Rt and inductance Lt . In order to test the capability of the EKF in tracking the total resistance and inductance under parameter variations, the value of the total resistance Rt is increased by 50% (as shown in Fig. 8). The EKF can successfully track the variation in the total resistance Rt . Moreover, there is only a very small effect on the estimation of the total inductance Lt or on the control performance of the FS-MPC scheme (see tracking accuracy of udc , idf , iqf in Fig. 8). Fig. 9 shows the performance of the proposed control system with EKF, when the total inductance Lt is increased by 50%. The EKF can again successfully follow the variation of the total inductance Lt with small coupling effect on the total resistance Rt . Moreover, the FS-MPC performance is still good (see Fig. 9). The proposed FS-MPC with on-line parameter estimation overcomes the problem of parameter uncertainty in current/power control of AFE converters. In order to illustrate the effectiveness of the proposed FS-MPC scheme with on-line parameter estimation, a simulation is performed where the total inductance is increased by 50% but the parameter estimation algorithm is disabled. Hence, the FS-MPC scheme will not obtain estimated (updated) model parameters and will work with wrong system parameters. Fig. 10

idf ,ref i df

i f

30 15

iˆf

iqf ,ref i qf

i f

iˆf

0 -15 -30

0.3

R t [ ]

Lt [mH ]

Rt Rˆ

ein

0.1 0.0 15

rank

ess

td

Q

al Co py

R t [ ]

0.3

P

5

udc [V ]

i f

10

i df & i qf [ A ]

0 -15 -30

P[kW] & Q[k var]

30 15

0.2

0.1

0.0

Rt Rˆ

t

15

Lt [mH ]

i f & i f [ A]

8

10

5

Lt Lˆ t

0

50

100 time [ms]

150

180

Fig. 7: Simulation results for step changes in the active P and reactive Q power (plotted quantities from top to bottom): Output active P and reactive Q power, reference udc,ref and actual udc DC-link voltage, reference idf,ref , iqf,ref and actual idf , iqf currents (in d/q), actual iα f, β α ˆβ ˆ if and estimated if , if currents (in α/β), total Rt and ˆ t resistance, and total Lt and estimated L ˆt estimated R

inductance.

shows the control performance for the DC-link voltage set-point tracking problem without parameter estimation. The DC-link controller is close to instability. The DC-link voltage oscillates around the set-point. This is because the reference currents reach their the maximally admissible values (i.e. imax = 35 A). The currents are at their (positive) limits and the model-based prediction is not correct anymore. The severe parameter mismatch (in Lt ) results in a wrong prediction of the future system behavior and, hence, an incorrect selection of switching states. The overall FS-MPC performance without on-line parameter is significantly deteriorated and close to instability (see Fig. 10). In order to check the noise filtering capability of the EKF, white noise is added to the current measurement (see Fig. 11 top). The EKF still works properly and does not amplify the measurement noise. The estimated cur-

Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters 0.3

Rt Rˆt

R t [ ]

R t [ ]

0.4 0.3 0.2 0.1 0.0

0.2

t

0.0

u dc

700

i df & i qf [ A]

698 25 20 15 10 5 0 -5

idf ,ref

40

60

i qf ,ref

i df

80 time [ms]

100

i qf

120

Fig. 8: Simulation results for a 50% step change in the total resistance Rt with on-line parameter estimation (plotted quantities from top to bottom): Total Rt and ˆ t resistance, total Lt and estimated L ˆ t inestimated R ductance, reference udc,ref and actual udc DC-link voltage, and reference idf,ref , iqf,ref and actual idf , iqf currents (in d/q).

R t [ ]

0.3 0.2

Rt Rˆ

0.1

t

0.0

u dc [V ]

Lt [mH ]

20 15

Lt Lˆ t

10 5 0 702

u dc , ref

700

25

5 -5

u dc

Fin

i df & i qf [ A ]

698

idf ,ref

40

60

Lt [mH ]

10 5 0 710

80 time [ms]

i qf ,ref

i df

100

i qf

u dc

690 50 40 30 20 10 0

i df ,ref

40

60

80 time [ms]

i df

100

i qf , ref

i qf

120

Fig. 10: Simulation results for a 50% step change in the total inductance Lt without on-line parameter estimation (plotted quantities from top to bottom): total ˆ t resistance, total Lt and estimated Rt and estimated R ˆ t inductance, reference udc,ref and actual udc DC-link L voltage, and reference idf,ref , iqf,ref and actual idf , iqf currents (in d/q).

The final simulation result is shown in Fig. 12 illustrating the impact of two different discretization techniques (forward Euler and Taylor series expansion) on the simulation accuracy. According to Fig. 12, the effect of the discretization method on the performance of the FS-MPC is small (since Ts = 40µs is chosen sufficiently small). The required time for the computation of the optimal switching vector using FS-MPC with forward Euler discretization is 5 µs, whereas FS-MPC with Taylor series expansion discretization requires 12 µs. Therefore, the simple forward Euler method is considered to be better suited for real-time implementation in this paper (and most of the related literature).

120

Fig. 9: Simulation results for a 50% step change in the total inductance Lt with on-line parameter estimation (plotted quantities from top to bottom): total Rt and ˆ t resistance, total Lt and estimated L ˆ t inestimated R ductance, reference udc,ref and actual udc DC-link voltage, and reference idf,ref , iqf,ref and actual idf , iqf currents (in d/q).

ˆβ rents ˆiα f and if have less noise content than the actual β measurements iα f and if (see Fig. 11 bottom).

u dc , ref

Lt Lˆ t

700

i df & i qf [ A ]

udc[V ]

u dc , ref

u dc [V ]

0 702

20 15

al Co py

Lt [mH ]

Lt Lˆ t

5

Rt Rˆ

0.1

15 10

15

9

6 Conclusion This paper proposed a finite set model predictive control (FS-MPC) scheme with on-line parameter estimation for current control of active front-end converters to overcome the model parameter uncertainty problem. The proposed on-line parameter estimation utilizes an extended Kalman filter (EKF) to estimate the filter impedance in addition to the grid impedance. Furthermore, the EKF is beneficial for filtering out noise in the current measurement. The observability matrix of the linearized system model is computed and the observability is checked on-line. Moreover, the delay time

M. Abdelrahem, C. Hack, and R. Kennel 3 0 -3 30 15

i f

0 -15 -30 0.0

20

iˆ f

i f

iˆ f

40 time [ms]

60

80

Fig. 11: Simulation results with additional white noise in the current measurement (plotted quantities from β top to bottom): white noise, and actual iα f , if and estimated ˆiα , ˆiβ currents (in α/β). f

f

udc [V ]

701 698

u dcEu

i df & i qf [ A]

695 25 20 15 10 5 0

Ty u dc

q idf ,Eu idf ,Ty i f ,Eu

0

30

4. P. Cortes, and et al., Predictive Control in Power Electronics and Drives, IEEE Transactions on Industrial Electronics, Vol. 55, No. 12, Dec. 2008, pp. 4312–4324. 5. A. Linder, R. Kanchan, R. Kennel, and P. Stolze, ModelBased Predictive Control of Electric Drives, G¨ ottingen, Germany: Cuvillier Verlag, 2010. 6. S. Kouro, and et al., Model Predictive Control–A Simple and Powerful Method to Control Power Converters, IEEE Transactions on Industrial Electronics, Vol. 56, No. 6, June 2009, pp. 1826–1838. 7. J. Rodriguez and P. Cortes, Predictive Control of Power Converters and Electrical Drives, 1st edition New York: Wiley-IEEE Press, 2012. 8. J. Rodriguez, and et al., State of the Art of Finite Control Set Model Predictive Control in Power Electronics, IEEE Transactions on Industrial Informatics, Vol. 9, No. 2, May 2013, pp. 1003–1016. 9. S. Vazquez, et al., Model Predictive Control: A Review of Its Applications in Power Electronics, IEEE Industrial Electronics Magazine, Vol. 8, No. 1, March 2014, pp. 16–31. 10. M. Abdelrahem, C. Hackl, and R. Kennel, Model Predictive Control of Permanent Magnet Synchronous Generators in Variable-Speed Wind Turbine Systems, in Proceedings of Power and Energy Student Summit (PESS 2016), Aachen, Germany, 19-20 January 2016. 11. M. Abdelrahem, C. Hackl, R. Kennel, Simplified Model Predictive Current Control without Mechanical Sensors for Variable-Speed Wind Energy Conversion Systems, Electrical Engineering Journal, vol. 99, no. 1, pp. 367-377, March 2017. 12. M. Abdelrahem, C. Hackl, and R. Kennel, Encoderless Model Predictive Control of Doubly-Fed Induction Generators in Variable-Speed Wind Turbine Systems, in Proceedings of The Science of Making Torque from Wind (TORQUE 2016) Conference, Munich, 5–7 October 2016. 13. M. H. Mobarak, M. Abdelrahem, N. Stati, and R. Kennel, Model predictive control for low-voltage ride-through capability improvement of variable-speed wind energy conversion systems, in proceedings of IEEE International Symposium on Industrial Electronics (INDEL), Banja Luka, 2016, pp. 1-6. 14. K. A. Islam, M. Abdelrahem, and R. Kennel, Efficient finite control set-model predictive control for grid-connected photovoltaic inverters, in proceedings of IEEE International Symposium on Industrial Electronics (INDEL), Banja Luka, 2016, pp. 1-6. 15. P. Cortes, J. Rodriguez, P. Antoniewicz, and M. Kazmierkowski, Direct power control of an AFE using predictive control, IEEE Transactions on Power Electron, Vol. 23, No. 5, pp. 2516–2523, 2008. 16. D. Quevedo, and et al., Model predictive control of an AFE rectifier with dynamic references, IEEE Transactions on Power Electronics, Vol. 27, No. 7, 2012, pp. 3128–3136. 17. M. Perez, R. Fuentes, J. Rodriguez, Predictive control of DC-link voltage in an active-front-end rectifier, in IEEE International Symposium on Industrial Electronics (ISIE), 27-30 June 2011, pp.1811–1816. 18. M. Perez, M. Vasquez, J. Rodriguez, J. Pontt, FPGAbased predictive current control of a three-phase active front end rectifier, in IEEE International Conference on Industrial Technology (ICIT), 10-13 Feb. 2009, pp.1–6. 19. S. Muslem Uddin, et al., Model predictive control of an active front end rectifier with unity displacement factor, in IEEE International Conference on Circuits and Systems (ICCAS), 18-19 Sept. 2013, pp.81–85.

al Co py

i f & i f [ A]

Noise[ A]

10

60 time [ms]

90

udc,ref

i qf ,Ty

120

Fig. 12: Simulation results using Forward Euler (Eu) and Taylor (Ty) series discretization method (plotted quantities from top to bottom): DC-link voltage udc , and actual idf , iqf currents (in d/q).

Fin

due to the digital calculation is compensated. The simulation results show that the proposed FS-MPC with on-line parameter estimation is capable of accurate and fast reference current tracking even under large variations in the model resistance and inductance. Furthermore, the EKF is capable of tracking the parameter variations with high accuracy and fast performance. For future work, the experimental implementation of the proposed FS-MPC with on-line parameter estimation is planned. Furthermore, a comparison between the adopted EKF and other techniques for noise rejection such as conventional low-pass filtering will be considered. References

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Finite Set Model Predictive Control with On-line Parameter Estimation for Active Frond-End Converters

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