Discrete-Time Sliding Mode Direct Power Control for Grid Connected Inverter with Comparative Study

Senad Huseinbegović, Branislava Peruničić-Draženović and Nijaz Hadžimejlić Department of automatic control and electronics

Faculty of Electrical Engineering, University of Sarajevo Sarajevo, Bosnia & Herzegovina

shuseinbegovic@etf.unsa.ba

Abstract — This paper presents a design of a digital direct power control strategy for a three-phase grid-connected inverter combining the discrete-time sliding mode control and the space vector modulation. Using the discrete-time state-space model of the controlled system, a discrete-time sliding mode control system is designed. Its output is a control vector which minimizes the instantaneous active and reactive power displacement from their reference values. The control vector is computed from the samples of voltages and currents and then converted to a switching sequence using space vector modulation. The period of modulated signal is equal to the sample period. A correction of the control vector is defined with aim to eliminate the influence of the system uncertainties using predicted values of the active and reactive powers. In this way a robust control system with a constant switching frequency is designed. Its digital hardware implementation is very simple. This control system is tested on a simulation model and compared with other similar approaches.

Keywords – direct power control, discrete-time sliding mode, robustness, grid connected inverter, space vector modulation.

I. INTRODUCTION The grid connected inverters are widely used in the modern

power systems [1]. In contemporary applications, the Direct Power Control (DPC), firstly proposed in [2], is recognized as the most promising strategy. The control vector of the inverter is computed using instantaneous active and reactive power values. Key features of this control strategy are a fast response and a good tracking capability. General classification of the DPC strategy is based on the constant switching frequency versus a variable switching frequency implementation [3].

In this paper, a DPC approach which operates with a constant switching frequency is adopted because this offers many advantages [3]. Up to now, presented approaches with a constant switching frequency may be classified into two groups. The DPC approaches where the constant switching frequency is achieved by Space Vector Modulation (SVM) are presented in [4-8]. Using SVM, a switching sequence is generated from a control vector which may be calculated using various control methods [3-8]. The second group contains the DPC approaches based on the predictive control, but without a modulator [3,9-12]. The switching sequence is generated by minimizing a cost function of the desired performance of the control system. It requires complicated online calculation, and that requires a more complex digital hardware.

A grid connected inverter may be seen as a variable structure system. Thus, it is a natural candidate for Sliding

Mode Control (SMC) application. It is known that SMC in a continuous-time (CT) system is robust with respect to matching external disturbance and parameter changes [13]. However, the variable, and even theoretically infinite, switching frequency was the main disadvantage of the SMC. The first work with a constant switching frequency SMC for power rectifier application are presented in the paper by Silva [14]. Afterwards, this approach has been applied in many papers. The SMC based DPC approaches are described in [3,7]. An interesting SMC DPC approach is presented in [7], where SMC block directly calculates the desired control vector, which eliminates the actual errors of active and reactive powers.

Very small sampling period that occur in CT-SMC is not acceptable for conventional digital hardware implementation. Besides, the inverter must also switch the state with a high frequency, and therefore the switching losses may be substantial. Thus, a control strategy with a switching time reduction can significantly improve the efficiency of a power inverter. However, an implementation with a low switching time may result in a chattering around the system trajectory, and even the instability of the control system. The discrete-time SMC (DT-SMC) intrinsically works with a constant frequency and it looks as a good approach, especially if the digital hardware is used in control system [13]. Also, DT-SMC allows a lower chattering frequency. The pioneering paper on the application of the DT-SMC approach in the power inverters was [15]. First of the important DT-SMC DPC with constant switching frequency approach was presented in [8]. However, the increasing sampling time enhances effect of the system uncertainties on the accuracy of the discrete-time control system. This drawback was not considered in any of the above cited approaches.

In this paper, the authors present a new digital DPC strategy for the grid connected inverter which combines DT-SMC and SVM. The control vector is obtained by using the equivalent control method [16]. The proposed strategy directly regulates the instantaneous active and reactive power injected into grid, and it can reasonably reject system uncertainties. Using a prediction method, the influence of the system uncertainties are eliminated by introducing the correction of the control vector. The proposed approach can operate with a low switching frequency. Using the test system implemented in MATLAB’s Simulink, the some advantages of proposed approach are demonstrated, analyzed and compared with approaches published in [4,7,8].

978-1-4799-2399-1/14/$31.00 ©2014 IEEE 459

II. MODEL OF THE GRID CONNECTE

A typical configuration of a three-phasinverter is shown in Fig. 1. Dynamic modsystem in the stationary αβ reference frame is

where: is the vector of the current injected i, is the vectors of the inverter and the is the grid filter resistance, is the grid filter inductance, is the grid angular frequency.

In (1), the voltage vector is related

vector of the inverter. Applied transformationreference frame, the voltage vector is given

, where is the control vector, and is DCeach time instant, the control vector is eqfeasible switching vectors of a three phase inIn above equations, the vectors , decomposed as T, lie in the plane.

Using these vectors, the instantaneous aand reactive power injected into tcalculated from voltage vector and curras:

Here represents the actual power vectorThe complete state-space model of a grid-cconvenient for DPC strategy is defined by (1), (2) and (3). All strategies presented obtained using this state-space model.

Fig. 1 Three-phase grid-connected inv

ED INVERTER se grid-connected del of the control s defined by: , (1)

into the grid, grid voltages,

with the control n in stationary αβ n as:

(2)

C-link voltage. In qual to one of the nverter topologies.

, and are . All vectors

active power the grid can be rents vector ,

. (3)

r in the plane. connected inverter

matrix equations in this paper are

III. RELEVANT PREVIOUSLY

Three DPC approaches wibecause it is essential for this be compared to the digital DPNote that a constant switchingin all approaches. It is one otechniques because of a highoutput voltage harmonics, smanot the least, it is suitaimplementation.

A. Convetional PI based DPCConventional control syste

control in a rotating referenreference current vector is comvector. The reference currentactual current vector. Their controllers which eliminate sfrom PI controllers make treference frame. This approacdigital implementation with a Its major disadvantage is a nthat it needs a complex hardwa

B. Continual-time SMC baseIn the paper [7], the authors

nonlinear SMC designed usincontrol system calculates the coreference frame in order to elimand reactive powers. The contr

· ·In previous equation, the m

as , 0In (4), the vector

between the reference and tsliding hyper-surface is defin

In above expressions, gains. The key advantage of threference frame, the phase angflux are not required.

C. Discrete-time SMC based A nonlinear DT-SMC DP

induction generator, using thepresented in [8]. The controlvector in the stationary αβ refeerrors of active and reactive pinduction generator. For thecontrol vector is defined as:

verter

Y PUBLISHED DPC APPROACHES ll be discussed in this section, research. These approaches will C strategy offered in this paper.

g frequency is achieved by SVM of the most popular modulation her DC-link utilization, smaller aller switching losses. Last, but

able for a digital hardware

C [4] em uses the closed-loop current nce frame. The value of the

mputed using the reference power t vector is compared with the

differences are inputs to PI steady-state errors. The outputs the control vector in rotating ch is convenient for analog or very high sampling frequency.

need for two transformations so are implementation.

d DPC [7] s propose a DPC approach using ng the Lyapunov function The ontrol vector in the stationary αβ minate the actual errors of active rol vector is defined by:

· · · .(4)

matrices , and are defined 0 and

00 .

represents the difference the actual power vectors. The ned by:

. (5)

, , and are control his approach is that the rotating gle of grid voltage or the virtual

DPC [8] PC approach for the double fed e Lyapunov stability concept is l system calculates the control erence frame that eliminates the powers on the stator side of the e grid connected inverter, the

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· · · · · , (6)

where is| | 00 . The

parameters and are positive constants which satisfy the conditions 1 and . The sliding hyper-surface is defined by the error vector between the reference and the instantaneous power vector. This provides the similar advantages as in the previous cited approach. The main disadvantage is the application of the Euler’s first order approximation in the discretization process which introduces errors if the sampling frequency is low.

IV. PROPOSED DISCRETE-TIME SLIDING MODE DPC In digital control power applications, the voltage and

current signals pass through A/D converters and therefore are available only at specific time instants. Also, the control signals can only be changed at same time instants. If in design process a continuous-time approach is translated into a discrete time approach using some approximation of derivatives, the discretization effect may be neglected only if the frequency sampling is high enough. If the sampling period is increased, the robustness of such control system is reduced. For example, the calculation of the reactive power from the voltage and current samples is very sensitive to the discretization effects. In this section, a new DT-SMC approach is proposed. Our design takes into consideration the system uncertainties: the sampling period, the quantization effects, the unknown time delay, the system parameter variation and non-idealities in digital SVM implementation.

A. Discrete Model of Grid Connected Inverter The first step in design of a discrete-time control system is

to make a discrete-time model of the controlled system. Starting from continuous-time state-space model equations (1), (2) and (3), and assuming zero-order hold on the control vector

, the discrete-time state-space model can be obtained as:

1 · · · , (7)

· , (8)

where are , 1 and 1 ( is 2 2 identity matrix). In (7)

and (8), is equivalent to time instant · , where is the sampling period. The discrete time model given by (7) and (8) is used to design DPC strategy in this paper.

B. Discrete-Time Sliding Mode DPC For DT-SMC DPC, the control task is that the actual power

vector reaches the value of the reference power vector starting from any feasible initial conditions in a finite time, and then to keep the vector representing the error between the reference power vector and the actual power vector at zero in the

steady state. Let the sliding hyper-surface in the time instant be defined as:

, (9)

where are and . The control task may be represented with equations 1 0 having as solution for control vector of the system given by (7) and (8) the equivalent control vector defined by:

, (10)

where cos sinsin cos . This matrix is

obtained from 1 using well known trigonometric equations.

The vector (10) brings the value of actual power vector to its reference value in just one sampling instant and then maintains the actual power vector on it for all the following sampling instants. However, the control vector (10) may not be available. Its Euclid norm is inversely proportional to the sampling time, and it may be very high if there is a big distance between the initial state and the sliding hyper-surface (9). To overcome this drawback, we use a control dependent on the distance of state from sliding hyper-surface named reaching law control [17, 18]. The control vector is defined as:

. (11)

In (11), the function represents a nonlinear function of the tracking errors and the switching functions. Depending on the selection of the function , the various control algorithms were proposed [17]. In this paper, the Bartolini’s control algorithm [17,18] is used. In this way, the magnitude of the control vector is limited by √2. The control vector (11) is obtained in the stationary reference frame, and can be directly transferred into the switching sequence by SVM. The modulation period is equal to the sampling period .

C. Suppression of the system uncertainties influence In the previous section, the main objective of the control

system was defined so that the actual power vector should track its reference value. Based on this, the equivalent control vector was obtained. But, if this control vector is applied, the tracking error does not converge to zero in the steady state if the system uncertainties are present. Therefore, a correction of the control vector is necessary to minimize the influence of the system uncertainties.

In this paper, the control vector will be corrected using the difference between the estimated power vector and the measured power vector. In each time instant, the difference between these two power vectors may be computed using values of their magnitudes and phase angle. Assuming that the amplitudes of harmonics of the voltage vector and current vector are very small with respect to their fundamental harmonic, the estimated power vector in time instant is:

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where is the Euclid norm of angle is the angle between the grid voltage predicted current vector . The vector is c

In reality, the estimated and actual powerin phase angle and magnitude. Let be the dphase angle of the actual and estimated powebe the ratio between amplitudes of actual andvector. Then the actual power vector destimated power vector value may be written

In steady-state mode DT-SMC DPC apprthe previously section, will ensure that the vto its reference vector , if the applied contrby (10). If the vector replaces the vectvector will not track the reference vector reference value defined by (13). Luckily the

can be calculated in each time instanindependently and will be comhardware implementation. The expressions ar

,

Now, it can be concluded that the actuawill track the reference vector , if the evector (10) is corrected using and and (15), respectively. This correction reference power vector in (10) should be corrected reference power vector defined b

In this way, while the estimated power vcorrected reference power vector , the actuwill track the reference values , and thesystem uncertainties will be rejected.

The proposed approach requires the coexpressions (14) and (15). In transient mode,lead to instability of the control system so ththe control vector should be applied in the only. The condition for its activation can be d

|1 sin δ cos δ |where should be small.

, (12)

the vector . The vector and the

computed by (7).

r vector may differ difference between er vector, and let d estimated power epending on the as:

. (13)

roach, proposed in vector converges rol vector is given tor in (13), the

, but the wrong parameters and

nt. Instead of , mputed for simple re defined by:

(14)

. (15)

al power vector equivalent control

defined by (14) implies that the replaced with the

by:

. (16)

vector tracks the ual power vector e influence of the

omputation of the the updating may hat a correction of steady-state mode

defined as: , (17)

From this viewpoint, the coobtained using expressions fopowers. This improves the system uncertainties rejection, system dynamic.

V. SIMUL

The simulation results of implemented in MATLAB’s Sdiagram of the system modesystem are listed in Table I. Usvector and sampling frequeincorporated into intervals operiod is divided into five timepower vectors (Table II). ThekHz and 6.4 kHz, are used dFigs 3 and 4, the instantaneouspower vector are shown for ewell as corresponding samplinapproaches apparently give analysis show that there is scenarios.

TABLE I. Parameters of the si

Symbol QuantiVdc the input DC voltage Vrms the nominal line to linωr the angular frequency R the resistance value L the inductance value ∆T the simulation step timT the total simulation tim

TABLE II. Reference powers for th

No Time interval [s] Int1 0.04, 0.10Int2 0.10, 0.16Int3 0.16, 0.22 Int4 0.22, 0.28Int5 0.28, 0.34

Fig. 2 MATLAB/Simulink block

orrection of the control vector is for average active and reactive

steady-state accuracy and the without any degradation of the

LATION RESULTS described DPC approaches are

Simulink. Fig 2. shows the block el. The electrical parameters of sing the various reference power encies, several scenarios are

of the simulation period. This e intervals with various reference e two sampling frequencies, 3.2 during computer simulation. In s power vector and the reference each of analyzed approaches, as ng frequency. Although all the satisfactory results, a detailed difference at handling various

imulated grid connected inverter

ity Value 800 V

e grids voltage 560 V of voltage 314 rad

500 mΩ 22 mH

me 1us me 0.36s

he simulated grid connected inverter

10.00 0.00 10.00 1.00 8.00 1.00 8.00 2.00 0.00 2.00

k diagram of the proposed strategy

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These differences are expressed through commonly

accepted figures of merit. It is difficult to say which approach is best, since there is yet no commonly accepted criterion. The comparison of behavior in mentioned time intervals is conducted for each simulated interval with respect to the ripple of the error powers, the mean error of the powers, the THD of the currents, the THD of the control vector and the response time.

For each figure of the merit, the data set is defined with members, where is number of approaches ( 4) and is the number of intervals ( 5). First, the members of data sets are normalized so that their value is divided by sum of all members of data for corresponding time interval. Then, the average rank of the each DPC approach is defined as the average value of the normalized values for corresponding approach.

(a) Conventional PI DPC approach

(b) CT-SMC DPC approach

(c) DT-SMC DPC approach

(d) Proposed DT-SMC DPC approach

Fig. 4. System response for the sampling frequency 6.4 kHz

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2

4

6

8

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12

Time [s]Act

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pow

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W] a

nd R

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VA

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2

4

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2

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pow

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r]

pP*

qQ*

(a) Conventional PI DPC approach

(b) CT-SMC DPC approach

(c) DT-SMC DPC approach

(d) Proposed DT-SMC DPC approach

Fig. 3. System response for the sampling frequency 3.2 kHz

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

2

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

0

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W] a

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2

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The ranking list of the compared approaches is presented in Table III. Total rank is computed as the Euclid norm of the ideal point (null vector). The approach with minimal total rank is assigned rank 1. Using previous table, the selection of DPC approach is possible so that satisfies corresponding figure of merit. For example, the mean power error is lowest by PI DPC approach, but this approach is unacceptable for other figures of merit. On the other hand, CT-SMC and DT-SMC are adequate for all figures of merit except for mean power error.

The proposed DPC approach has somewhere slower transient response than two other SMC approaches, but faster than PI DPC approach. The main advantage of this approach is that the mean power errors are minimized as in the PI DPC approach. Too, other performances are retained at the same level as in the SMC DPC approaches. Unlike other approaches, the proposed DPC approach does not have any worst ranking figure of merit. One of the important contributions is that our approach gives good results for low switching frequency.

VI. CONCLUSION In this paper, a new digital DPC approach for grid

connected inverter is proposed combing discrete-time SMC and SVM. In each sampling instant, the SMC maintenance conditions for proposed approach are defined using equivalent control method.

Using predictive control concept, the approach has possibility to minimize the system uncertainties. The correction of the control vector is accepted using the actual and estimated power values based on the sample of voltage and current.

The proposed approach requires only basic mathematical computation, and it is better suited for digital hardware implementation. For various switching frequency, the simulation results present the effectiveness and robustness of the proposed approach during variations of the reference active and reactive power.

The performances of the proposed approach are compared with the performances of other similar approaches, where the proposed digital DPC approach is given very good results especially for low switching frequency.

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[4] M. Malinowski, M. Jasinski and M.P. Kazmierkowski, “Simple direct power control of three-phase PWM rectifier using space-vector modulation (DPC-SVM),” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 447–454, Apr. 2004.

[5] J.A. Restrepo and et., “Optimum space vector computation technique for direct power control," IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1637-1645, Jun. 2009.

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[7] J. Hu, L. Shang, Y. He, and Z.Q. Zhu, “Direct active and reactive power regulation of grid-connected DC/AC converters using sliding mode control approach,” IEEE Trans. Power Electron., vol. 26, no. 1, pp. 210–222, Jan. 2011.

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[9] S. Aurtenechea, M.A. Rodriguez, E. Oyarbide and J.R. Torrealday, “Predictive control strategy for DC/AC converters based on direct power control,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1261–1271, Jun. 2007.

[10] P. Cortes, J. Rodriguez, P. Antoniewicz and M. Kazmierkowski, “Direct power control of an AFE using predictive control,” IEEE Trans. Power Electron., vol. 23, no. 5, pp. 2516–2523, Sep. 2008.

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[12] S.Huseinbegovic and B.Perunicic-Drazenovic, “Discrete-time sliding mode direct power control for three-phase grid connected multilevel inverter,” in Proc Int Conf. Power Engineering, Energy and Electrical Drives, pp. 933-938, May 2013

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[15] S.K. Mazumder, “A novel discrete control strategy for independent stabilization of parallel three-phase boost converters by combining SVM with Variable-Structure Control,” IEEE Trans. Power Electron., vol. 18, no. 4, pp.1070-1083., July 2003.

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[18] G. Bartolini, A. Ferrara, and V.I. Utkin, “Adaptive sliding mode control in discrete-time systems,” Automatica, vol. 31 (5), pp. 769-773, 1995.

TABLE III. Results of the comparative study

. Average rank .

Average rank A

ppro

ach

PI D

PC

CT

-SM

C D

PC

DT

-SM

C D

PC

Prop

osed

DPC

PI D

PC

CT

-SM

C D

PC

DT

-SM

C D

PC

Prop

osed

DPC

0.35 0.21 0.21 0.23 0.29 0.23 0.23 0.25 0.33 0.23 0.23 0.21 0.28 0.25 0.24 0.23 0.29 0.25 0.26 0.20 0.25 0.27 0.25 0.22 0.35 0.20 0.18 0.27 0.30 0.20 0.23 0.27 0.15 0.28 0.35 0.22 0.27 0.34 0.12 0.27 0.08 0.36 0.48 0.08 0.05 0.56 0.35 0.05

0.29 0.24 0.24 0.24 0.26 0.25 0.25 0.25 0.39 0.20 0.20 0.20 0.35 0.20 0.22 0.23 0.39 0.20 0.20 0.20 0.34 0.21 0.22 0.23

0.54 0.11 0.12 0.22 0.70 0.10 0.06 0.14 Total rank 1.08 0.74 0.84 0.67 1.09 0.90 0.72 0.71

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