Model Predictive Approach for a Simple and ... - Semantic Scholar
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014
5259
Model Predictive Approach for a Simple and Effective Load Voltage Control of Four-Leg Inverter With an Output LC Filter Venkata Yaramasu, Student Member, IEEE, Marco Rivera, Member, IEEE, Mehdi Narimani, Member, IEEE, Bin Wu, Fellow, IEEE, and José Rodriguez, Fellow, IEEE
ge s
to the loads in stand-alone power systems. The loads are of single-/three-phase, balanced/unbalanced, and linear/nonlinear nature [5]. The three-phase three-leg voltage source inverter (VSI) with an output inductive–capacitive (LC) filter is a simple structure, where the load neutral is connected to the midpoint of the dc-link capacitors [6]. The dc-link capacitors are overrated to withstand the load neutral current flowing through them. Moreover, the step change in load currents causes a surge, which will destroy the dc-link capacitors. The three-phase fourleg inverters are promising candidates for three-phase four-wire applications. In this configuration, the load neutral is connected to the fourth-leg instead of dc-link capacitor midpoint. The four-leg inverter provides enhanced dc bus utilization (15% higher compared to the three-leg inverter), lower ripple on the dc-link voltage, and smaller size for dc-link capacitors [7], [8]. The four-leg inverter with an output LC filter is used in many applications such as [3], [4], [9], [10] stand-alone DG [11], uninterruptible power supplies (UPSs) [12], and power quality equipment such as dynamic voltage restorers (DVRs) [13] and universal power quality conditioners (UPQCs) [14]. It has been reported in literature that a four-leg inverter with an output inductive (L) filter can be used in the grid-connected DG and shunt active power filter applications [2], [15]–[21]. In the applications that involve an LC filter, the output voltage is controlled, while the output current is regulated in L-filterbased applications. Many classical control techniques have been previously analyzed for the load voltage control. They include the following: hysteresis regulators [22], linear PID controllers [8], pole placement controllers [23], variable structure control [24], and sliding mode control [25]. All of these control methods (except hysteresis) produce modulating signals, and they are used by the modulation stage to generate the gating signals for a four-leg inverter [26]. The modulation schemes for the four-leg inverters can be broadly classified as the following: carrier less (hysteresis [27] and flux vector [28]) and carrier based (sinusoidal pulsewidth modulation (SPWM) [29], [30], selective harmonic elimination (SHE) based PWM [31], and 3-D space vector modulation (3D-SVM) [7], [12], [32]). The carrier-less modulation methods cause a variable switching frequency operation, while the carrier-based methods impose a constant switching frequency operation. The hysteresis and flux vector techniques use complicated switching tables. The
Pa
Abstract—This paper presents a finite control set model predictive strategy and its application to the load voltage control of two-level four-leg inverters. The proposed approach uses the novel discrete-time model of the inverter and output LC filter in order to predict the variables to be controlled. These predictions are carried out for the 16 switching states of the inverter and are evaluated using a cost function. The switching state that forces the load voltages to be closest to their respective references is chosen and applied to the inverter. The behavior of the predictive controller has been investigated, and the changes to both inductive and capacitive filter parameters have been considered. In order to improve the reliability of the fourth leg as well as the overall inverter efficiency, a solution is proposed, which combines hardware and software reconfigurations. The feasibility of the proposed method is verified through simulation and experimental results considering single-/three-phase, balanced/unbalanced, and linear/nonlinear loads.
I. I NTRODUCTION
S
e
Index Terms—DC–AC power conversion, digital control, distributed generation, four-leg inverters, predictive control.
Sa
m
pl
TAND-ALONE distribution generation (DG) systems have been receiving much attention recently for their advantages in supplying power to remote customers. They constitute the most promising alternative to the expensive and complicated grid connection [1]. A variety of stand-alone systems include the following: single home, large communities, islands, satellite stations, aircrafts, ship propulsion systems, and large-scale computer systems [2]–[4]. Power electronic converters play an important role in providing clean power
Manuscript received March 16, 2013; revised August 9, 2013 and October 17, 2013; accepted November 25, 2013. Date of publication January 2, 2014; date of current version May 2, 2014. This work was supported by the Natural Sciences and Engineering Research Council of Canada through Wind Energy Strategic Network (WESNet) Project 3.1, by Fondecyt Initiation into Research 11121492, by CONICYT PIIC 2048 Project, and by the Universidad Técnica Federico Santa María. This work was made possible by NPRP Grant 4-077-2-028 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors. V. Yaramasu, M. Narimani, and B. Wu are with the Department of Electrical and Computer Engineering, Ryerson University, Toronto ON M5B 2K3, Canada (e-mail: [email protected]; [email protected]; bwu@ ee.ryerson.ca). M. Rivera is with the Department of Industrial Technologies, Universidad de Talca, Merced 437, Chile (e-mail: [email protected]). J. Rodriguez is with the Electronics Engineering Department, Universidad Técnica Federico Santa María, Valparaíso 110-V, Chile (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2297291
0278-0046 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014
Fig. 1. Two-level four-leg inverter topology with an output LC filter and three-phase arbitrary load.
the gap in this research area. The main contributions of this work are summarized as follows. 1) The concept of FCS-MPC is extended to the load voltage control of two-level four-leg inverters with single-/threephase, unsymmetrical, and nonlinear loads. 2) A novel discrete-time model of the system with the output LC filter and neutral inductor Ln is presented. 3) The robustness of the proposed controller is studied with both the inductive and capacitive filter parameter variations. 4) A new software-based neutral-leg switching frequency reduction algorithm is also proposed. This solution is integrated with the hardware reconfiguration to achieve better results. In this paper, simulations are carried out using MATLAB/ Simulink software, and the experimental results are obtained using a dSPACE DS1103 controller. This paper is organized as follows. In Section II, the continuous and discrete-time models of the system are presented; this is followed by the explanation of the proposed control strategy in Section III. The practical issues in the digital implementation are discussed in Section IV. The simulation and experimental results are presented together in Section V. The conclusion is presented in Section VI.
Sa
m
pl
e
Pa
SHE-PWM is mainly based on the switching angle calculation, and its real-time implementation is complicated. The 3D-SVM is most favorable compared to the SPWM as it offers good dc bus utilization as well as a good load harmonic profile. However, this method is complex in nature, and thus, its software and digital implementation are complicated. The finite control set model predictive control (FCS-MPC) has recently emerged as a new control tool in power electronics and drives. This scheme offers a conceptually different approach to control the power converters. Similar to the carrierless modulation schemes, this method operates with the variable switching frequency. However, this method is based on a relatively simple concept, and it provides an intuitive approach for the real-time implementation as well as a fast dynamic response. Moreover, the predictive control approach is flexible; thus, the additional constraints such as switching frequency minimization, spectrum shaping, common-mode voltage minimization, and switching/conduction loss minimization can be included in the design of the controller [33]–[40]. Since model predictive control (MPC) uses the model of the system, as the name implies, it is important to establish an accurate system model. This requirement has prompted many studies, which are detailed in [2]–[4], [6], [9], [10], [15]–[21], and [41]–[43]. These works can be broadly classified into two groups based on their applications; they involve either output current control [2], [15]–[21], [41]–[43] or voltage control [3], [4], [6], [9], [10]. A predictive current control strategy with an output L filter is presented in [43] for a two-level three-leg VSI. This concept has been extended to four-leg two-level [2], [15]–[19] and three-level [most popularly known as neutral-point clamped (NPC)] [20] VSIs. To improve the reliability of the neutral leg, the works in [15], [18], and [19] proposed hardwarebased solutions (neutral inductor Ln ), while the works in [16] and [17] proposed software-based solutions (cost function penalization). The robustness of a predictive controller with L filter parameter variations is studied in [16] and [17]. The application of a two-level four-leg inverter for active power filtering is analyzed in [18] and [19], where a fast dynamic response has been demonstrated in comparison to the classical controllers. With this feature, nearly sinusoidal currents have been obtained on the source side despite the highly distorted currents on the load side. In [21], the authors have combined the predictive control with space vector modulation to achieve fixed switching frequency operation; however, this approach is too complex and nonintuitive for real-time implementation. The works in [41] and [42] extended the concept of predictive current control to the four-leg indirect matrix converter. On the other hand, the predictive load voltage control of a two-level three-leg VSI is analyzed in [6] considering symmetrical loads. A preliminary simulation study is carried out for the load voltage control of two-level [3], [4], [9], and NPC [10] four-leg inverters. However, these works use the same load model introduced in [6] for the three-wire applications. Considering the research done in this area to date and the lack of system models available for the four-wire applications with output LC filter and neutral inductor Ln , this work aims to fill
ge s
5260
II. M ODEL OF THE S YSTEM The four-leg inverter topology with an output LC filter is shown in Fig. 1. The generation and storage units of stand-alone power systems are replaced by a constant dc voltage source. The midpoint of the fourth leg is connected to the LC filter and load through a neutral inductor Ln . The fourth leg controls the load zero sequence voltage [5]. The neutral inductor Ln helps in minimizing the switching frequency ripple imposed on the neutral inverter current (in ). The inverter and load neutrals are represented by n and o, respectively. To simplify the analysis, the following voltage and current vectors are defined: v = [vun
vvn
vwn ]T
(1)
vo = [vou
vov
vow ]T
(2)
T
(3)
i = [iu io = [iou
iv iov
iw ]
iow ]T .
(4)
YARAMASU et al.: MODEL PREDICTIVE APPROACH FOR A SIMPLE AND EFFECTIVE LOAD VOLTAGE CONTROL
5261
The ac-side voltages of the four-leg inverter can be given in terms of switching signals and dc-side voltage as [15] vyn = vyN − vnN = (Sy − Sn )vdc
y = u, v, w
(5)
where vdc and vnN are the dc-link and load neutral voltages, respectively. By applying Kirchhoff voltage and current laws to the nodes given in Fig. 1, the following load dynamics can be obtained: di d in + v − R n in − L n (6) dt dt d vo (7) i = io + Cf dt in = − (iu + iv + iw ) ion = −(iou + iov + iow ). (8)
ge s
vo = R f i + Lf
where
I 0 0 Cf B = −1 −1 −L −Leq Req 6×6 L−1 eq ⎤ ⎡ eq Rf + Rn Rn Rn = ⎣ Rn Rf + Rn Rn ⎦ Rn Rn Rf + Rn ⎤ ⎡ Ln Ln Lf + Ln = ⎣ Ln Lf + Ln Ln ⎦ Ln Ln Lf + Ln
Req
Leq
−I Cf
0
6×6
Fig. 2. Block diagram of the proposed predictive voltage control scheme.
The cost function is defined such that it deals with the control objectives. In this paper, two control objectives are chosen and defined as follows:
(10)
g(k + 1) = [vo∗ (k + 1) − vo (k + 1)]2
e
A=
Pa
By solving (6)–(8), the inductive and capacitive filter models can be expressed in state-space form as vo v d vo =A +B (9) io dt i i
m
Sa where Ts is the sampling time.
III. D ESCRIPTION OF P REDICTIVE VOLTAGE C ONTROL The block diagram of the proposed load voltage control approach based on the predictive strategy is shown in Fig. 2. One-step prediction horizon [44] is considered in this paper to simplify the analysis and to reduce the computational burden. The control strategy requires the computation of the 16 possible conditions in sampling instant (k), which would minimize a given cost function in the (k + 1) sampling instant. The switching state that gives minimal value for the cost function is chosen and applied by the system during all of the (k + 1) period.
(13)
where λswc is the weighting factor. The first term in (13) deals with the reference tracking of load voltages, which is the main objective of this control algorithm. The generation of references is a different issue, and it depends primarily on the specific application. In stand-alone DG systems, the voltage references are generated based on the load appliance rating, and the magnitudes and frequencies can have different values on three phases. In DVR and UPQC applications, the source voltages should be maintained as sinusoidal, and the references are calculated such that they compensate the source voltage sags [13], [14]. To simplify the analysis and generalize the proposed method, the references are specifically provided by the user. In (13), vo∗ (k + 1) is the extrapolated reference obtained from vo∗ (k). The guidelines given in [15]–[17] can be used for the extrapolation. vo (k + 1) in (13) is the predicted load voltage. The 16 inverter switching combinations and the measured dc-link voltage vdc (k) are used for the prediction of the inverter voltages vun (k), vvn (k), and vwn (k) [refer to (5)]. These predictions, along with the measured variables vo (k), i(k), and io (k), are used to estimate the future behavior of load voltages vo (k + 1) [refer to (11)]. The second term in (13) is defined to reduce the neutralleg switching frequency. The load-neutral voltage vnN is minimized by the fourth leg. To accomplish this, the fourth leg operates at a higher switching frequency in comparison to the other three phases. This scenario leads to an increase in the switching losses of the fourth leg [16]. The neutral inductor
pl
where 0 and I are third-order null and identity matrices, respectively. The mathematical models in [3], [4], [6], and [9] are presented without considering the neutral inductor Ln . The model presented here is useful for feeding single-phase/threephase and unbalanced/nonlinear loads. This model can be used with other converter configurations such as NPC and matrix converters. The digital implementation of the proposed control algorithm requires discrete-time model. The continuous-time state-space system in (9) can be converted to discrete form as vo (k) v(k) vo (k + 1) (11) =Φ +Γ i(k + 1) i(k) io (k) Φ = eATs , Γ = A−1 (Φ − I6×6 )B (12)
+λswc | Sn (k + 1) − Sn,opt (k) |
5259
Model Predictive Approach for a Simple and Effective Load Voltage Control of Four-Leg Inverter With an Output LC Filter Venkata Yaramasu, Student Member, IEEE, Marco Rivera, Member, IEEE, Mehdi Narimani, Member, IEEE, Bin Wu, Fellow, IEEE, and José Rodriguez, Fellow, IEEE
ge s
to the loads in stand-alone power systems. The loads are of single-/three-phase, balanced/unbalanced, and linear/nonlinear nature [5]. The three-phase three-leg voltage source inverter (VSI) with an output inductive–capacitive (LC) filter is a simple structure, where the load neutral is connected to the midpoint of the dc-link capacitors [6]. The dc-link capacitors are overrated to withstand the load neutral current flowing through them. Moreover, the step change in load currents causes a surge, which will destroy the dc-link capacitors. The three-phase fourleg inverters are promising candidates for three-phase four-wire applications. In this configuration, the load neutral is connected to the fourth-leg instead of dc-link capacitor midpoint. The four-leg inverter provides enhanced dc bus utilization (15% higher compared to the three-leg inverter), lower ripple on the dc-link voltage, and smaller size for dc-link capacitors [7], [8]. The four-leg inverter with an output LC filter is used in many applications such as [3], [4], [9], [10] stand-alone DG [11], uninterruptible power supplies (UPSs) [12], and power quality equipment such as dynamic voltage restorers (DVRs) [13] and universal power quality conditioners (UPQCs) [14]. It has been reported in literature that a four-leg inverter with an output inductive (L) filter can be used in the grid-connected DG and shunt active power filter applications [2], [15]–[21]. In the applications that involve an LC filter, the output voltage is controlled, while the output current is regulated in L-filterbased applications. Many classical control techniques have been previously analyzed for the load voltage control. They include the following: hysteresis regulators [22], linear PID controllers [8], pole placement controllers [23], variable structure control [24], and sliding mode control [25]. All of these control methods (except hysteresis) produce modulating signals, and they are used by the modulation stage to generate the gating signals for a four-leg inverter [26]. The modulation schemes for the four-leg inverters can be broadly classified as the following: carrier less (hysteresis [27] and flux vector [28]) and carrier based (sinusoidal pulsewidth modulation (SPWM) [29], [30], selective harmonic elimination (SHE) based PWM [31], and 3-D space vector modulation (3D-SVM) [7], [12], [32]). The carrier-less modulation methods cause a variable switching frequency operation, while the carrier-based methods impose a constant switching frequency operation. The hysteresis and flux vector techniques use complicated switching tables. The
Pa
Abstract—This paper presents a finite control set model predictive strategy and its application to the load voltage control of two-level four-leg inverters. The proposed approach uses the novel discrete-time model of the inverter and output LC filter in order to predict the variables to be controlled. These predictions are carried out for the 16 switching states of the inverter and are evaluated using a cost function. The switching state that forces the load voltages to be closest to their respective references is chosen and applied to the inverter. The behavior of the predictive controller has been investigated, and the changes to both inductive and capacitive filter parameters have been considered. In order to improve the reliability of the fourth leg as well as the overall inverter efficiency, a solution is proposed, which combines hardware and software reconfigurations. The feasibility of the proposed method is verified through simulation and experimental results considering single-/three-phase, balanced/unbalanced, and linear/nonlinear loads.
I. I NTRODUCTION
S
e
Index Terms—DC–AC power conversion, digital control, distributed generation, four-leg inverters, predictive control.
Sa
m
pl
TAND-ALONE distribution generation (DG) systems have been receiving much attention recently for their advantages in supplying power to remote customers. They constitute the most promising alternative to the expensive and complicated grid connection [1]. A variety of stand-alone systems include the following: single home, large communities, islands, satellite stations, aircrafts, ship propulsion systems, and large-scale computer systems [2]–[4]. Power electronic converters play an important role in providing clean power
Manuscript received March 16, 2013; revised August 9, 2013 and October 17, 2013; accepted November 25, 2013. Date of publication January 2, 2014; date of current version May 2, 2014. This work was supported by the Natural Sciences and Engineering Research Council of Canada through Wind Energy Strategic Network (WESNet) Project 3.1, by Fondecyt Initiation into Research 11121492, by CONICYT PIIC 2048 Project, and by the Universidad Técnica Federico Santa María. This work was made possible by NPRP Grant 4-077-2-028 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors. V. Yaramasu, M. Narimani, and B. Wu are with the Department of Electrical and Computer Engineering, Ryerson University, Toronto ON M5B 2K3, Canada (e-mail: [email protected]; [email protected]; bwu@ ee.ryerson.ca). M. Rivera is with the Department of Industrial Technologies, Universidad de Talca, Merced 437, Chile (e-mail: [email protected]). J. Rodriguez is with the Electronics Engineering Department, Universidad Técnica Federico Santa María, Valparaíso 110-V, Chile (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2297291
0278-0046 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014
Fig. 1. Two-level four-leg inverter topology with an output LC filter and three-phase arbitrary load.
the gap in this research area. The main contributions of this work are summarized as follows. 1) The concept of FCS-MPC is extended to the load voltage control of two-level four-leg inverters with single-/threephase, unsymmetrical, and nonlinear loads. 2) A novel discrete-time model of the system with the output LC filter and neutral inductor Ln is presented. 3) The robustness of the proposed controller is studied with both the inductive and capacitive filter parameter variations. 4) A new software-based neutral-leg switching frequency reduction algorithm is also proposed. This solution is integrated with the hardware reconfiguration to achieve better results. In this paper, simulations are carried out using MATLAB/ Simulink software, and the experimental results are obtained using a dSPACE DS1103 controller. This paper is organized as follows. In Section II, the continuous and discrete-time models of the system are presented; this is followed by the explanation of the proposed control strategy in Section III. The practical issues in the digital implementation are discussed in Section IV. The simulation and experimental results are presented together in Section V. The conclusion is presented in Section VI.
Sa
m
pl
e
Pa
SHE-PWM is mainly based on the switching angle calculation, and its real-time implementation is complicated. The 3D-SVM is most favorable compared to the SPWM as it offers good dc bus utilization as well as a good load harmonic profile. However, this method is complex in nature, and thus, its software and digital implementation are complicated. The finite control set model predictive control (FCS-MPC) has recently emerged as a new control tool in power electronics and drives. This scheme offers a conceptually different approach to control the power converters. Similar to the carrierless modulation schemes, this method operates with the variable switching frequency. However, this method is based on a relatively simple concept, and it provides an intuitive approach for the real-time implementation as well as a fast dynamic response. Moreover, the predictive control approach is flexible; thus, the additional constraints such as switching frequency minimization, spectrum shaping, common-mode voltage minimization, and switching/conduction loss minimization can be included in the design of the controller [33]–[40]. Since model predictive control (MPC) uses the model of the system, as the name implies, it is important to establish an accurate system model. This requirement has prompted many studies, which are detailed in [2]–[4], [6], [9], [10], [15]–[21], and [41]–[43]. These works can be broadly classified into two groups based on their applications; they involve either output current control [2], [15]–[21], [41]–[43] or voltage control [3], [4], [6], [9], [10]. A predictive current control strategy with an output L filter is presented in [43] for a two-level three-leg VSI. This concept has been extended to four-leg two-level [2], [15]–[19] and three-level [most popularly known as neutral-point clamped (NPC)] [20] VSIs. To improve the reliability of the neutral leg, the works in [15], [18], and [19] proposed hardwarebased solutions (neutral inductor Ln ), while the works in [16] and [17] proposed software-based solutions (cost function penalization). The robustness of a predictive controller with L filter parameter variations is studied in [16] and [17]. The application of a two-level four-leg inverter for active power filtering is analyzed in [18] and [19], where a fast dynamic response has been demonstrated in comparison to the classical controllers. With this feature, nearly sinusoidal currents have been obtained on the source side despite the highly distorted currents on the load side. In [21], the authors have combined the predictive control with space vector modulation to achieve fixed switching frequency operation; however, this approach is too complex and nonintuitive for real-time implementation. The works in [41] and [42] extended the concept of predictive current control to the four-leg indirect matrix converter. On the other hand, the predictive load voltage control of a two-level three-leg VSI is analyzed in [6] considering symmetrical loads. A preliminary simulation study is carried out for the load voltage control of two-level [3], [4], [9], and NPC [10] four-leg inverters. However, these works use the same load model introduced in [6] for the three-wire applications. Considering the research done in this area to date and the lack of system models available for the four-wire applications with output LC filter and neutral inductor Ln , this work aims to fill
ge s
5260
II. M ODEL OF THE S YSTEM The four-leg inverter topology with an output LC filter is shown in Fig. 1. The generation and storage units of stand-alone power systems are replaced by a constant dc voltage source. The midpoint of the fourth leg is connected to the LC filter and load through a neutral inductor Ln . The fourth leg controls the load zero sequence voltage [5]. The neutral inductor Ln helps in minimizing the switching frequency ripple imposed on the neutral inverter current (in ). The inverter and load neutrals are represented by n and o, respectively. To simplify the analysis, the following voltage and current vectors are defined: v = [vun
vvn
vwn ]T
(1)
vo = [vou
vov
vow ]T
(2)
T
(3)
i = [iu io = [iou
iv iov
iw ]
iow ]T .
(4)
YARAMASU et al.: MODEL PREDICTIVE APPROACH FOR A SIMPLE AND EFFECTIVE LOAD VOLTAGE CONTROL
5261
The ac-side voltages of the four-leg inverter can be given in terms of switching signals and dc-side voltage as [15] vyn = vyN − vnN = (Sy − Sn )vdc
y = u, v, w
(5)
where vdc and vnN are the dc-link and load neutral voltages, respectively. By applying Kirchhoff voltage and current laws to the nodes given in Fig. 1, the following load dynamics can be obtained: di d in + v − R n in − L n (6) dt dt d vo (7) i = io + Cf dt in = − (iu + iv + iw ) ion = −(iou + iov + iow ). (8)
ge s
vo = R f i + Lf
where
I 0 0 Cf B = −1 −1 −L −Leq Req 6×6 L−1 eq ⎤ ⎡ eq Rf + Rn Rn Rn = ⎣ Rn Rf + Rn Rn ⎦ Rn Rn Rf + Rn ⎤ ⎡ Ln Ln Lf + Ln = ⎣ Ln Lf + Ln Ln ⎦ Ln Ln Lf + Ln
Req
Leq
−I Cf
0
6×6
Fig. 2. Block diagram of the proposed predictive voltage control scheme.
The cost function is defined such that it deals with the control objectives. In this paper, two control objectives are chosen and defined as follows:
(10)
g(k + 1) = [vo∗ (k + 1) − vo (k + 1)]2
e
A=
Pa
By solving (6)–(8), the inductive and capacitive filter models can be expressed in state-space form as vo v d vo =A +B (9) io dt i i
m
Sa where Ts is the sampling time.
III. D ESCRIPTION OF P REDICTIVE VOLTAGE C ONTROL The block diagram of the proposed load voltage control approach based on the predictive strategy is shown in Fig. 2. One-step prediction horizon [44] is considered in this paper to simplify the analysis and to reduce the computational burden. The control strategy requires the computation of the 16 possible conditions in sampling instant (k), which would minimize a given cost function in the (k + 1) sampling instant. The switching state that gives minimal value for the cost function is chosen and applied by the system during all of the (k + 1) period.
(13)
where λswc is the weighting factor. The first term in (13) deals with the reference tracking of load voltages, which is the main objective of this control algorithm. The generation of references is a different issue, and it depends primarily on the specific application. In stand-alone DG systems, the voltage references are generated based on the load appliance rating, and the magnitudes and frequencies can have different values on three phases. In DVR and UPQC applications, the source voltages should be maintained as sinusoidal, and the references are calculated such that they compensate the source voltage sags [13], [14]. To simplify the analysis and generalize the proposed method, the references are specifically provided by the user. In (13), vo∗ (k + 1) is the extrapolated reference obtained from vo∗ (k). The guidelines given in [15]–[17] can be used for the extrapolation. vo (k + 1) in (13) is the predicted load voltage. The 16 inverter switching combinations and the measured dc-link voltage vdc (k) are used for the prediction of the inverter voltages vun (k), vvn (k), and vwn (k) [refer to (5)]. These predictions, along with the measured variables vo (k), i(k), and io (k), are used to estimate the future behavior of load voltages vo (k + 1) [refer to (11)]. The second term in (13) is defined to reduce the neutralleg switching frequency. The load-neutral voltage vnN is minimized by the fourth leg. To accomplish this, the fourth leg operates at a higher switching frequency in comparison to the other three phases. This scenario leads to an increase in the switching losses of the fourth leg [16]. The neutral inductor
pl
where 0 and I are third-order null and identity matrices, respectively. The mathematical models in [3], [4], [6], and [9] are presented without considering the neutral inductor Ln . The model presented here is useful for feeding single-phase/threephase and unbalanced/nonlinear loads. This model can be used with other converter configurations such as NPC and matrix converters. The digital implementation of the proposed control algorithm requires discrete-time model. The continuous-time state-space system in (9) can be converted to discrete form as vo (k) v(k) vo (k + 1) (11) =Φ +Γ i(k + 1) i(k) io (k) Φ = eATs , Γ = A−1 (Φ − I6×6 )B (12)
+λswc | Sn (k + 1) − Sn,opt (k) |
