Predictive optimal switching sequence direct power control for grid connected power converters

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1 Predictive Optimal Switching Sequence Direct Power Control for Grid Connected Power Converters Sergio Vazquez, Member,IEEE, Abraham Marquez, Ricardo Aguilera, Member,IEEE, Daniel Quevedo, Senior Member,IEEE, Jose I. Leon, Senior Member,IEEE and Leopoldo G. Franquelo, Fellow,IEEE Abstract—Grid connected power converters play a key role in several applications such as integration of renewable energy sources and motor drives. For this reason, the development of high performance control strategies for this particular class of power converters has increasingly attracted the interest of both academic and industry researchers. This paper presents the predictive Optimal Switching Sequence Direct Power Control (OSS-DPC) algorithm for grid connected converters. The OSSDPC method belongs to the predictive direct power control (P-DPC) family and provides the desired power references by calculating globally optimal switching sequences. To address computational and implementation issues, an efficient control algorithm, named reduced OSS-DPC (ROSS-DPC) is introduced. The implementation of the proposed control strategy in a standard DSP is evaluated on a two-level power converter prototype working as a STATCOM. Experimental results show algorithm’s potential to provide high performance during both transient and steady states. I. INTRODUCTION GRID connected power converters play a key role in many industrial applications such as active front end (AFE) converters for motor drives, integration of renewable energy sources (wind or photovoltaic), and energy storage systems [1]–[4]. To achieve high performance, several issues have been studied in the literature, including converter topologies, modulation strategies, grid synchronization schemes, and control algorithms [5]–[10]. The control scheme of a grid connected converter is usually divided into two control loops. The external one is devoted to regulate the dc-link capacitor voltage. The inner control loop focuses on either tracking the current or instantaneous active and reactive power references depending on the state variables Manuscript received January 29, 2014; revised March 25, 2014, May 9, 2014 and July 29, 2014; accepted July 29, 2014. Copyright c 2014 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported by the Ministerio Espa˜nol de Economia y Competitividad under project ENE2012-36897 and by the Consejeria de Innovaci´on Ciencia y Empresa (Junta de Andalucia) under the project P11- TIC-7070. S. Vazquez, A. Marquez, J.I Leon and L.G. Franquelo are with the Electronic Engineering Department, Universidad de Sevilla (Spain), (e-mail: ser[email protected]). Ricardo Aguilera is with the Australian Energy Research Institute, The University of New South Wales, Australia ([email protected]). Daniel Quevedo is with the School of Electrical Engineering and Computer Science, The University of Newcastle, Australia ([email protected]). used to design the control strategy. In both cases, it is possible to use an indirect control approach by including a PWM-SVM modulation scheme [11], [12]. The direct power control (DPC) method uses the instantaneous active and reactive powers as state variables. In its conventional form, DPC directly selects the power switch states to track the desired active and reactive powers [13]. The main problem with this approach is that the resulting switching frequency is variable. This complicates the design of the output filter. To overcome this issue, DPC formulated as an indirect control was proposed in [14]. In order to avoid performance degradation by the mismatching of system parameters, an adaptive DPC control law has also been developed [15]. Although [14], [15] perform well, two major issues remain. First, the method to define the control design constants is not always clear and usually requires a trial and error procedure. Second, conventional PI controllers or integral structures are used to derive the control action. These factors limit the bandwidth of the DPC algorithm, sacrificing the dynamic response of the system. As an alternative to these approaches, model predictive control (MPC) has been applied to the design of DPC strategies (MPC-DPC). In general, MPC solves, at each sampling instant, an optimal control problem over a finite prediction horizon. This optimization yields an optimal control sequence for the whole prediction horizon. Following the receding horizon principle, only the first control action of this optimal sequence is applied to the system. This process is repeated at each sampling instant using fresh state measurements or estimates [16]–[19]. Depending on the nature of the input constraints, MPC can be divided into two major groups. If the power converter uses a modulator then the input will be constrained to belong to a bounded continuous control set (e.g., duty cycles or PWM references). In the case when the converter switches are handled directly, the input will belong to a Finite Control Set (FCS) [18]. In the power electronics literature, both continuous and FCS-MPC algorithms have been proposed for the DPC of power converters [20]–[23]. To highlight this difference, in the present work they will be referred to as P-DPC and FCS-P- DPC respectively. FCS-P-DPC considers the converter switches as the system input. A power cost function, which comprises future active and reactive power tracking errors is evaluated for each IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2 possible switch combination. The optimal input to be applied to the system is the one that minimizes this cost function. FCS-P-DPC is able to provide good performance. However, it requires high sampling frequency and provides a variable switching frequency. This generates output currents with a widespread harmonic spectrum [20], [21]. Alternatively, P-DPC considers the commutation instants ti, within a sampling period Ts, as the system input. To do this, for each sampling period, P-DPC selects a converter voltage vector sequence from a table using the grid voltage vector angle. Thus, the minimization of the power cost function returns the optimal commutation instants top ifor these vectors within the sampling period. This generates a switching pattern that provides a constant switching frequency [24]. The robustness of this strategy can be improved by using observers to identify uncertain system parameters such as the smoothing inductor value [25], [26]. Conventional P-DPC achieves, in general, good closed-loop behavior. Nonetheless, at times, its performance will be affected by an incorrect voltage sequence selection [27]. To solve this problem, revised tables defining the optimal output voltage sequence have been proposed in [28]. In essence, when a switching time becomes negative, then a new voltage sequence is selected. Although this procedure presents good results, it requires additional computations. To overcome this issue, a P-DPC strategy, which requires neither grid voltage sector information nor a voltage vector selection process, was proposed in [29]. In this strategy, the controller simply evaluates the switching times for one sector and, from this information, reconstructs the optimal switching pattern to be applied. The associated constrained optimization problem has also been addressed in [30]. It should be noted that natural constraints for this problem are the switching period and the magnitude of the control voltage vector. The solution to the constrained problem proposed in [31], consists of calculating the global optimum from the quadratic cost function of the instantaneous power deviations at the end of the predicted switching instant. Thus, the control vector is chosen as the orthogonal projection of the global optimum to the hexagon of the control vectors. The present paper introduces a novel P-DPC strategy. To overcome the aforementioned problems inherent to voltage sequence selection, the use of look-up-tables is avoided. Instead, the proposed P-DPC takes advantage of the discrete nature of the power converter. As is well known, the control region of the power converter in the stationary αβ frame can be divided into six sectors. Taking this into account, the proposed controller calculates the optimal commutation instants for each sector. This provides six local Optimal Switching Sequences (OSSs) and their associated cost values. Similar to the FCSMPC approach, the global OSS applied by the proposed P- DPC is that which minimizes the power cost function. The main advantage of the proposed predictive controller when compared to P-DPC is that the control action is calculated without relying on the grid voltage sector information. This allows the controller to select the OSS even when the control action is saturated. The proposed predictive control strategy is called OSS-DPC. n van vbn vcn ia ib ic Fig. 1. Scheme of a grid connected two-level power converter. The present article extends the preliminary work presented in [32]. In the latter, only a simulation study was carried out. In the present work, the proposed OSS-DPC is experimentally tested in order to evaluate the effectiveness of the proposed strategy when implemented on a standard DSP platform. It turns out that OSS-DPC requires a large number of computations. To address this issue, a new algorithm, named Reduced OSS-DPC (ROSS-DPC) is developed. ROSS-DPC presents a reduced computational cost when compared to OSS-DPC, allowing one to take advantage of the digital platform to improve the closed-loop performance. In addition, in this paper the effect of delays caused by the digital implementation are also investigated and a delay compensator is introduced. The remainder of this paper is organized as follows. In Section II the conventional P-DPC controller is described. In Section III the proposed OSS-DPC is derived. Experimental results are documented in Section IV. Here, in order to address practical experimental issues, the ROSS-DPC algorithm is introduced. Conclusions are given in Section V. II. CONVENTIONAL P-DPC ALGORITHM The key idea of the conventional P-DPC strategy is to use a predefined switching pattern based on the available switching vectors that the converter can generate. In order to minimize the active and reactive power errors, the method finds, in an optimal manner, the instants when these vectors are applied within a given sampling period. For this purpose, this section develops a system model which is then used to analyze the instantaneous active and reactive power dynamics [22], [24]. A. System Model Fig. 1 shows the electrical scheme of a voltage source inverter (VSI) connected to the grid. Considering the stationary αβ framework, the instantaneous active and reactive power, ps and qs, can be expressed by [33]: ps qs=vsα vsβ vsβ −vsαisα isβ ,(1) where vsαβ and isαβ represent the grid voltage and the VSI output current in the αβ frame respectively by using the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 3 vr1 vr0 vr2 vr3 vr4 vr5vr6 vr7® ¯ Sector 1 Sector 2 Sector 12 Sector 11 Sector 3 Sector 10 vs®¯ Fig. 2. Converter voltage vectors vri. power-invariant Clarke transformation. From (1), it is possible to derive dynamic equations for psand qsas dps dt =vsα disα dt +vsβ disβ dt +isα dvsα dt +isβ dvsβ dt , dqs dt =vsβ disα dt −vsα disβ dt −isβ dvsα dt +isα dvsβ dt . (2) Under a balanced sinusoidal three-phase system, the derivative of the grid voltage can be expressed as dvsαβ dt =ωJvsαβ,(3) where the matrix Jis given by J=0−1 1 0 .(4) Taking into account that the dynamic model of the ac currents can be represented by disαβ dt =1 L(vrαβ −vsαβ ),(5) where vrαβ is the output voltage of the VSI, the following dynamic model of the active and reactive powers is obtained dps dt =vsα 1 L(vrα −vsα) + ωisβ +vsβ 1 L(vrβ −vsβ )−ωisα, dqs dt =vsβ 1 L(vrα −vsα) + ωisβ −vsα 1 L(vrβ −vsβ)−ωisα. (6) B. Vector Selection As illustrated in Fig. 2 a two-level power converter can generate eight voltage vectors, vri with i∈ {0,...,7}in the αβ plane. In the conventional P-DPC [22], the vector sequence applied to the converter is selected in order to TABLE I VECTOR SEQUENCE TO BE APPLIED WHEN USING CONVENTIONAL P-DPC Sector Vector sequence ~vj 1 1 2 7 7 2 1 2 0 1 2 2 1 0 3 0 3 2 2 3 0 4 3 2 7 7 2 3 5 3 4 7 7 4 3 6 0 3 4 4 3 0 7 0 5 4 4 5 0 8 5 4 7 7 4 5 9 5 6 7 7 6 5 10 0 5 6 6 5 0 11 0 1 6 6 1 0 12 1 6 7 7 6 1 minimize the switching losses. To do this, the αβ plane is divided into twelve sectors as depicted in Fig. 2. Each sector is assigned to a vector sequence, ~vj={v1, v2, v3, v3, v2, v1} with j∈ {1,...,12}as presented in Table I. The sequence to be applied to the converter during the sampling period, Ts, is chosen depending on the sector in which the grid voltage lies. For example, in Fig. 2, vsαβ lies inside sector 1, thus the sequence to be applied is ~v1={vr1, vr2, vr7, vr7, vr2, vr1}. C. Application Times Once the switching pattern is selected, the commutation instants t1,t2, and t3for the three voltage vectors viwith i∈ {1,2,3}that form the switching sequence ~vjneed to be calculated. The P-DPC approach adopted in [22] is based on the dynamic model presented in (6). To be more specific, the predictions for the active and reactive powers at instant k+ 1 are calculated as ps,k+1 =ps,0+ 2 3 X i=1 fpiti qs,k+1 =qs,0+ 2 3 X i=1 fqiti, (7) where ps,k+1 and qs,k+1 are the predictions for instant k+ 1, ps,0and qs,0are the initial values of psand qsat instant k, fpi,fqi are the increments for the instantaneous active and reactive powers, and tiare the application times of vectors vri. If it is assumed that fpi and fqi are constant during the sampling interval, then fpi =fpi,k and fpi =fqi,k, where fpi,k and fqi,k are the values at instant kfor the vector vi. These can be calculated from (6) as fpi,k =dps dt vr=vi,vs=vs,k,is=is,k fqi,k =dqs dt vr=vi,vs=vs,k,is=is,k . (8) To obtain the commutation instants ti, the conventional P- DPC minimizes the quadratic function J(ps,k+1, qs,k+1) = e2 p,k+1 +e2 q,k+1 (9) IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 4 vr1 vr0 vr2 vr7® ¯ Sector1 Sector2 Sector12 Sector3 v s vr ± v svr P Q, ssXL XL I s Ù V s Ù V r Ù vr1 vr0 vr2 vr7® ¯ Sector1 Sector2 Sector12 Sector3 v s vr ± XL I s Ù V s Ù V r Ù isis b) c) a) Fig. 3. Vectorial representation of the steady state. a) Simplified system. b) Required inverter voltage. c) Constrained inverter voltage using conventional P-DPC with ep,k+1 =p∗ s−ps,k+1 and eq,k+1 =q∗ s−qs,k+1, where p∗ sand q∗ sare the reference values for the instantaneous active and reactive power respectively. In [22], an explicit solution to the above problem is found by considering an unconstrained problem. This yields the following application times t1=(fq2−fq3)epk + (fp3−fp2)eqk + (fp2fq3−fp3fq2)Ts 2 ((fq2−fq3)fp1+ (fq3−fq1)fp2+ (fq1−fq2)fp3), t2=(fq3−fq1)epk + (fp1−fp3)eqk + (fq1fp3−fq3fp1)Ts 2 ((fq2−fq3)fp1+ (fq3−fq1)fp2+ (fq1−fq2)fp3), t3=Ts 2−t1−t2, (10) where ep,k =p∗ s−ps,0,eq,k =q∗ s−qs,0and Tsrepresents the sampling period. III. OSS-DPC TECHNIQUE Conventional P-DPC, as shown in Section II, constitutes a simple strategy that often provides good performance when controlling grid connected power converters. However, at times, the voltage sequence selection method used by P- DPC will perform poorly [27]. The vectorial representation presented in Fig. 3 illustrates this problem. In essence, vr should be generated using the voltage vectors belonging to the sector where vsis located. Then, at times, it is not possible to generate the required voltage to achieve the instantaneous active and reactive power references [32]. To overcome this problem, [32] proposed an algorithm to optimally obtain the sector in which the inverter voltage, vr, will be synthesized. The key idea of this method, named OSSDPC, is to calculate the local optimal times in (10), for every sector, i.e., Tj= [ t1, t2, t3]for all j∈ {1,...,12}. These times are used to evaluate, in each sector j, the cost function Jj(t1, t2, t3), i=6 X i=1 (p∗ s−psi)2+ (q∗ s−qsi)2,(11) qs0 ps0 ps1 ps2 ps3 ps4 ps5 ps6 qs1 qs2 qs3qs4 qs5qs6 fq1 fq2fq3fq1 fq2 fq3 fp1 fp2fp3 fp1 fp2 fp3 t1t2t3t1 t2 t3 Ts ps, qs t kk+1 Fig. 4. Active and reactive power trajectories during one sampling period. j=1 ? J= f f, pi qi i²{ }1,2,3 vj Table 1 (Sector) 1 i²{ }1,...,6 T t t t=[ , , ] 2 3 Eq. (10) J<* J j j12 j j= +1 * J=J j yes yes no T T= op * no T T= * j j ~ v v= ~op ~ * * v ~vj ~ = ps0, qs0 psi, qsi Eq. (12) Eq.(8) Fig. 5. Flow diagram of the proposed OSS-DPC strategy. where psi and qsi are defined as psi =ps(i−1) +fpiti qsi =qs(i−1) +fqiti,(12) IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 5 0.145 0.150 0.155 0.160 −2000 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s) qs [VAr] qs ? qs 0.140 (a) 0.145 0.150 0.155 0.160 −2000 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (s) qs [VAr] qs ? qs 0.140 (b) Fig. 6. System response to an instantaneous reactive power command step from q∗ s= 0 kVAr to q∗ s= 10 kVAr: (a) P-DPC algorithm, (b) OSS-DPC algorithm. 0.26 0.27 0.28 0.29 0.3 −30 −20 −10 0 10 20 30 Time (s) ia, ib, ic [A] ia ib ic (a) 0.26 0.27 0.28 0.29 0.3 −30 −20 −10 0 10 20 30 Time (s) ia, ib, ic [A] ia ib ic (b) Fig. 7. Steady state grid currents for instantaneous reactive power reference q∗ s= 10 kVAr: (a) P-DPC algorithm, (b) OSS-DPC algorithm. with ps0=ps,0,qs0=qs,0,t4=t3,t5=t2,t6=t1, fp4=fp3,fp5=fp2and fp6=fp1. Fig. 4 illustrates the evolution of psand qsfrom instant kto k+ 1 The twelve resulting cost values, Jj, are local minima. The optimal sector, jop, is chosen as the one which minimizes the cost function in (11). The optimal times for this sector are Top, whereas ~vop is the global optimal switching sequence. In Fig. 5, a flow diagram of the OSS-DPC algorithm is presented. The main advantage of using (11) when compared to (9) is that it evaluates the predictions for the instantaneous active and reactive powers at six different instants within the sampling period. This improves the inter-sample behavior, cf. [34]. To show the benefits of using the OSS-DPC, a simulation has been carried out using the parameters presented in Table II. Fig. 6 compares the response to a step reference change in TABLE II SYSTEM PARAMETERS Parameter Value Smoothing inductor L= 2 mH DC-Link capacitor C= 2200 µF Grid phase voltage van, vbn, vcn = 230 Vrms Grid frequency f= 50 Hz Sampling frequency fs= 10 kHz Switching frequency fsw = 10 kHz Fig. 8. Converter prototype. q∗ susing conventional P-DPC and the proposed OSS-DPC algorithms. Fig. 6a and Fig. 6b show that both strategies present similar dynamic responses, although a slightly faster response can be observed when OSS-DPC is used. However, OSS-DPC offers a clear improvement when compared with standard P-DPC in the steady state. To highlight this, Fig. 7 presents the grid current waveforms. Fig. 7a illustrates that conventional P-DPC periodically selects an incorrect switching sequence. This generates spikes in the grid currents, which increase their total harmonic distortion (THD). On the other hand, the grid currents obtained by OSS-DPC do not have spikes, as shown in Fig. 7b. This is due to the fact that OSSDPC explores all vector sectors when obtaining the optimal switching sequence. IV. EXPERIMENTAL RESULTS The proposed OSS-DPC algorithm has been tested on a laboratory prototype system comprising a power converter connected to the grid. The experimental setup consists of a three-phase two-level converter working as a STATCOM. The electrical scheme of the system is shown in Fig. 1. The main system parameters are summarized in Table II. The converter prototype is shown in Fig. 8. The core of the control hardware is based on the TMS320F28335 DSP where the OSS-DPC algorithm has been coded. During the experiments, instantaneous reactive power command steps have been introduced in order to evaluate performance. Fig. 9 shows the transient response of the OSS-DPC when a command step is produced in the instantaneous reactive IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 6 0 0.02 0.04 0.06 0.08 0.1 −2000 0 2000 4000 6000 8000 10000 12000 Time (s) qs [VAr] qs qs ? Fig. 9. System response to an instantaneous reactive power command step from 0 kVAr to 10 kVAr. (a) (b) Fig. 10. Steady state output currents for an instantaneous reactive power command of 10 kVAr: (a) Output currents, (b) Harmonic content. power from no load to 10 kVAr. The system response is very fast and the new reference is reached almost instantaneously. However, in steady state, there exists a difference between the command and actual value of q. This is mainly caused by model inaccuracies, and the delay introduced by the digital platform. The latter issue can be compensated in the algorithm design as will be shown later in Section IV-B. The output currents of the VSI are shown in Fig. 10. It should be noted that the low order harmonics in the output currents are due to the grid voltage in the laboratory utility, which contains 5th and 7th harmonics. The THD value of the output current is 4.1 %. (a) (b) Fig. 11. Execution time of the control strategies: (a) OSS-DPC algorithm, (b) ROSS-DPC algorithm. ® ¯ V0 V7 A C E F D B vr1 vr2 vr3 vr4vr0 vr7 vr5vr6 Fig. 12. Sectors considered in the ROSS-DPC algorithm. A. Analysis of the computational burden One of the major concerns about MPC algorithms is the computational cost. In general, MPC strategies rely upon significant online computations and therefore require powerful hardware platforms for real time applications. To evaluate the computational burden of the proposed OSSDPC algorithm, its execution time has been measured. Fig. 11a shows graphically the execution time of the algorithm. To do this, one of the DSP pins is set to ’1’ when the internal interruption stars. Then it is toggled to ’0’ when the algorithm is finished. It can be observed that the time needed to perform all the calculations is roughly 90 µs. This shows that the proposed OSS-DPC strategy requires intensive computations. Taking into account that, in the experiment, the sampling period is 100 µs, it turns out that OSS-DPC uses 90 % of the available time. Thus, it is not possible to increase the sampling frequency in the selected hardware platform. Consequently, the control strategy cannot take advantage of PWM-SVM optimizations including its ability to duplicate the sampling frequency and maintain the switching frequency to increase the system performance without increasing the switching losses. In order to reduce the number of computations needed, the following observations can be made. In the αβ frame vr0= vr7. Therefore, evaluations of sectors 1 and 2 lead to the same control action value. The same conclusion can be obtained for sectors pairs 3-4, 5-6, 7-8, 9-10 and 11-12. Thus, the control algorithm only needs to be evaluated for six sectors, from A to F, as presented in Fig. 12. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 7 TABLE III PARAMETERS fpi AND fqi TO BE APPLIED WITH ROSS-DPC. Sector fp1fp2fp3fq1fq2fq3 Afp,vr0fp,vr1fp,vr2fq,vr0fq,vr1fq,vr2 Bfp,vr0fp,vr3fp,vr2fq,vr0fq,vr3fq,vr2 Cfp,vr0fp,vr3fp,vr4fq,vr0fq,vr3fq,vr4 Dfp,vr0fp,vr5fp,vr4fq,vr0fq,vr5fq,vr4 Efp,vr0fp,vr5fp,vr6fq,vr0fq,vr5fq,vr6 Ffp,vr0fp,vr1fp,vr6fq,vr0fq,vr1fq,vr6 ps0,qs0 i = 0 Calculate fp;vri;fq;vri Eq. (8) i < 6 i = i+1 yes j = A fpi;fqi , i∈{1,2,3} Table III no Calculate Tj = [t1;t2;t3] Eq. (10) Saturate Tj Calculate psi;qsi , i∈{1,...,6} Eq. (12) j < Fj = j+1 yes yes no no Fig. 13. Flow diagram of the proposed ROSS-DPC algorithm. On the other hand, vector vr1is common to sectors A and F. Similarly, vr2belongs to sectors A and B, vr3to sectors B and C, vr4to sectors C and D, vr5to sectors D and E, and vr6to sectors E and F. Further, vr0and v7are included in 0 0.02 0.04 0.06 0.08 0.1 −2000 0 2000 4000 6000 8000 10000 12000 Time (s) qs [VAr] qs qs ? (a) (b) Fig. 14. Experimental results for the ROSS-DPC algorithm: (a) System response for an instantaneous reactive power command step from 0 kVAr to 10 kVAr. (b) Harmonic content of the output currents. Fig. 15. Harmonic content of the output currents for the ROSS-DPC with fs = 20 kHz and fsw = 10 kHz. all sectors. In this way, the values fpi and fqi for vector vri, denoted as fp,vri and fq,vri respectively, can be calculated in advance to the switching times corresponding to each sector, thereby providing a more efficient algorithm. Once a sector is selected, Table III can be used to choose the corresponding values of fpi and fqi. The above considerations lead to the reduced OSS-DPC (ROSS-DPC) algorithm. The flow diagram of the proposed ROSS-DPC strategy is shown in Fig. 13. The execution time of the ROSS-DPC has also been assessed graphically as presented in Fig. 11b. which demonstrate that the new algorithm only needs roughly 40 µs for computing all calculations. Therefore, the computational burden of the OSS-DPC has been reduced IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 8 k k - 1 k + 1 k + 2 ... qs;k qs;k+1(vop,Top)qs;k+2(sector(A)) qs;k+2(sector(C)) qs;k+2(sector(E)) Fig. 16. Sampling delay compensation for the proposed predictive control strategy. 0 0.02 0.04 0.06 0.08 0.1 −2000 0 2000 4000 6000 8000 10000 12000 Time (s) qs [VAr] qs qs ? (a) 0 0.02 0.04 0.06 0.08 0.1 −2000 0 2000 4000 6000 8000 10000 12000 Time (s) qs [VAr] qs qs ? (b) Fig. 17. Steady state response for an instantaneous reactive power command of 10 kVAr: (a) ROSS-DPC algorithm without delay compensation. (b) ROSSDPC algorithm with delay compensation. by 55 %. It is important to emphasize that only the number of computations have been reduced. The performance remains the same as shown in Fig. 14. Both transient and steady state responses are equal to those exhibited by OSS-DPC. Due to the computational savings, it is possible to run the ROSS-DPC using a sampling frequency of fs= 20 kHz and a switching frequency of fsw = 10 kHz. This improves the THD of the output current without increasing the switching losses. The harmonic spectrum of the output currents is presented in Fig. 15. It should be noticed that the THD is 2.9 %. Thus, a reduction of about 30 % has been achieved by increasing the sampling frequency. B. Delay compensation The implementation of the control algorithm on a digital platform introduces a delay in the control action which deteriorates the closed-loop performance of the system. This is due Fig. 18. Harmonic content of the output currents for the ROSS-DPC with delay compensation, fs= 20 kHz and fsw = 10 kHz. to the fact that, to account for computation times, the control action is applied one sampling period after it is calculated. In the case of MPC strategies, this can be easily handled by considering predictions at instant k+ 2 instead of instant k+ 1, [35], [36]. This is shown in Fig. 16 for the instantaneous reactive power variable. The value of the instantaneous reactive power at instant k(qs,k) can be calculated from the measurements vsαβ,k and isαβ,k. The value of this variable at instant k+ 1 only depends on vsαβ,k and isαβ,k and the switching function applied to the power converter. This switching function was calculated in the previous sampling instant by the ROSS-DPC and is defined by ~vop and Top. Therefore, it is possible to obtain a prediction of qs,k+1. Predictions at instant k+2 are needed to compute the algorithm. It should be noted that qs,k+2 depends on the sector evaluated and therefore it is necessary to calculate predictions for the values fpi and fqi at k+ 1,fpi,k+1 and fqi,k+1 respectively. Taking into account the definitions for the instantaneous active and reactive power, fpi,k+1 and fqi,k+1 can be calculated by fpi,k+1 =vsα,k+1 L(vrα −vsα,k+1) +vsβ,k+1 L(vrβ −vsβ,k+1)(13) −ωqs,k+1 fqi,k+1 =vsβ,k+1 L(vrα −vsα,k+1) −vsα,k+1 L(vriβ −vsβ,k+1)(14) +ωqs,k+1. From (13) and (14), it is clear that predictions for vsαβ,k+1 are needed. These values, under input balanced grid voltage can be easily obtained as vsα,k+1 vsβ,k+1=cos (2πfTs)−sin (2πfTs) sin (2πfTs)cos (2πfTs)vsα,k vsβ,k,(15) where Ts= 1/fsis the sampling period. The effect of the delay compensation on the performance of the ROSS-DPC strategy has been experimentally tested. For this purpose, the steady state response of the system for an IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 9 instantaneous reactive power command of 10 kVAr has been analyzed. Fig. 17a and Fig. 17b present the steady state system response, without and with the delay compensation respectively. In both cases the controller behaves correctly. However, when the average value of qsis analyzed it shows that without delay compensation qs,av is 9.723 kVAr whereas qs,av with delay compensation is 9.909 kVAr. Therefore, the relative error, defined as 100∗(q∗ s−qs,av)/q∗ s, is reduced from 2.77 % to 0.91 %. The compensation of the delay also improves the harmonic spectrum of the output current as depicted in Fig. 18. It can be noticed that the THD has been further reduced to 2.8 %. V. CONCLUSION Direct power control (DPC) strategies are a good alternative to control grid connected converters. Model predictive control (MPC) for power converters and drives has shown its ability to provide highly effective solutions. In this way, the predictive DPC (P-DPC) was developed in the literature in order to take advantage of both DPC and MPC features. However, there are still some issues to be addressed. In this paper, a new P-DPC controller, named optimal switching sequence DPC (OSSDPC), is proposed, solving the problem of conventional P- DPC with the selection of the optimal sequence from the grid voltage sector information. The proposed OSS-DPC algorithm takes advantage of the discrete nature of the power converter by obtaining a local optimum for each of the six voltage sectors. Since only a finite number of local optima exist, the global optimum can be obtained in a similar fashion as in the FCS-MPC case. Experiments have been conducted and the results show that the proposed control method performs well. A careful analysis of OSS-DPC showed that it is possible to reduce the number of calculations needed to implement the control strategy without reducing its performance. Additionally, the effect of the delay introduced due to digital implementation was also studied. 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