An enhanced finite control set model predictive control method with self‐balancing capacitor voltages for three‐level T‐type rectifiers
2022; Institution of Engineering and Technology; Volume: 15; Issue: 6 Linguagem: Inglês
10.1049/pel2.12245
ISSN1755-4543
AutoresSertaç Bayhan, Hasan Kömürcügil, Naki Güler,
Tópico(s)Microgrid Control and Optimization
ResumoIET Power ElectronicsVolume 15, Issue 6 p. 504-514 ORIGINAL RESEARCHOpen Access An enhanced finite control set model predictive control method with self-balancing capacitor voltages for three-level T-type rectifiers Sertac Bayhan, Corresponding Author Sertac Bayhan [email protected] orcid.org/0000-0003-2027-532X Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha, Qatar Department of Electrical-Electronic Engineering, Technology Faculty, Gazi University, Ankara, 06500 Turkey Correspondence Sertac Bayhan, Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, 5825 Doha, Qatar. Email: [email protected]Search for more papers by this authorHasan Komurcugil, Hasan Komurcugil Department of Computer Engineering, Eastern Mediterranean University, Via Mersin 10, Famagusta, TurkeySearch for more papers by this authorNaki Guler, Naki Guler orcid.org/0000-0003-4145-4247 Technical Sciences Vocational School, Gazi University, Ankara, 06500 TurkeySearch for more papers by this author Sertac Bayhan, Corresponding Author Sertac Bayhan [email protected] orcid.org/0000-0003-2027-532X Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha, Qatar Department of Electrical-Electronic Engineering, Technology Faculty, Gazi University, Ankara, 06500 Turkey Correspondence Sertac Bayhan, Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, 5825 Doha, Qatar. Email: [email protected]Search for more papers by this authorHasan Komurcugil, Hasan Komurcugil Department of Computer Engineering, Eastern Mediterranean University, Via Mersin 10, Famagusta, TurkeySearch for more papers by this authorNaki Guler, Naki Guler orcid.org/0000-0003-4145-4247 Technical Sciences Vocational School, Gazi University, Ankara, 06500 TurkeySearch for more papers by this author First published: 24 January 2022 https://doi.org/10.1049/pel2.12245Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract An effective finite control set model predictive control (FCS-MPC) is introduced for single-phase three-level T-type rectifiers supplying resistive as well as constant power loads (CPL). The main problem of CPL is the negative resistance phenomenon that endangers the rectifier's stability. Hence, the proposed FCS-MPC method is based on Lyapunov's stability theory such that the stability of the rectifier is guaranteed under all operating points. Unlike the existing FCS-MPC methods, the cost function design in the proposed control method is formulated on the rectifier's stability. According to Lyapunov's stability theory, the rectifier stays stable provided that the rate of change of Lyapunov function is negative. In this case, the derivative of the Lyapunov function can be used as the cost function without utilizing any weighting factor. Therefore, contrary to the existing FCS-MPC methods, the weighting factor requirement is eliminated which leads to easiness in the design and implementation of the controller. Experimental results reveal that the proposed control approach exhibits very good performance with undistorted and distorted grid voltage conditions when the rectifier feeds resistive and CPL loads. 1 INTRODUCTION Over the last three decades, the rectifiers have been very attractive in many systems including motor drives [1], storage systems in transportation and grid applications [2], wind turbine systems [3], hybrid AC/DC energy systems [4] and electric vehicles [5]. Although conventional two-level rectifiers provide satisfactory performance in these systems, multilevel converters have emerged that offer prominent advantages in terms of efficiency and cost [6-8]. Also, when two-level and multilevel converters are operated with the same switching frequency, the multi-level converters can generate much smaller current harmonics than that of two-level converters. Thus, compared to the two-level converters, the multilevel converters are able to deliver the same voltage (or current) quality by utilizing fewer passive components. In the developed multilevel converter topologies, the three-level T-type converter offers significant features like decreased component count [7], capability to operate at faulty conditions [9] and reduced losses. These features are significant if high efficiency and reduced cost are needed in low voltage applications [10, 11]. For this reason, the three-level T-type rectifiers became popular recently [12-19]. To control power converters, many powerful nonlinear control techniques like sliding mode control (SMC) [20] and model predictive control (MPC) [21] are developed for power converters. However, the control of three-level T-type rectifiers is not studied extensively. In literature, the control of T-type rectifiers is reported in several papers [16-18]. In Khan et al. [16] and Bayhan and Komurcugil [17], SMC is introduced for T-type rectifiers. Although the SMC is insensitive to parameter variations, it is not preferred in real applications because of chattering. In Komurcugil et al. [18], a passivity-based control (PBC) with improved robustness is introduced for single-phase T-type rectifiers. However, the robustness can be ensured if the controller gain is selected appropriately. In Khan et al. [19], a finite control set model predictive control (FCS-MPC) approach with observer ability is introduced for single-phase T-type rectifiers [19]. However, this MPC approach requires two weighting factors in the cost function. Since there is no predefined selection criterion for the weighting factors, the optimum weighting factor values are generally selected by a trial-and-error approach. In this case, the design of optimum weighting factors becomes challenging and time consuming. Even though the weighting factor can be tuned for a specific operating point of the converter, the control performance may be deteriorated when there is a large variation in the operating point. The main reason for this deterioration is the weighting factor, which is not adjusted for the new operating point. Therefore, adaptive weighting factor tuning methods are essential [22]. Some unknown variables such as lower and higher limits are needed in many adaptive tuning strategies [23]. Also, these methods increase computational burden. Therefore, the weighting factorless MPC methods can be specified as the better solution compared with MPC methods employing adaptive weighting factor mechanism [24]. The control methods presented in previous studies [16-19] offer various advantages regarding dynamic performance, insensitivity to parameter variations and easiness in the practical application. However, the common disadvantage of these control methods is that the controller design is based on the resistive load only without considering the constant power loads (CPLs). This means that the effectiveness of these control approaches with CPL may be deteriorated due to the negative resistance phenomenon of CPL. When a CPL is connected to the rectifier, the constant power consumption can be accomplished if the load bus voltage is increased while the CPL current is decreased and vice versa [25, 26]. Therefore, since CPL endangers the rectifier's stability due to the negative resistance phenomenon [27, 28], the assurance of stability is extremely important when both resistive load and CPL are connected to the rectifier at the same time. Conventional linear control techniques cannot guarantee system's stability because of the negative impedance characteristic of CPL and the nonlinear nature of power converters [29, 30]. In this study, a FCS-MPC based on Lyapunov's stability theory is introduced for a single-phase T-type rectifier which not only feeds a resistive load, but also a constant power load. The Lyapunov's direct method utilizes a Lyapunov energy function. The derivative of the Lyapunov energy function should be negative so as to stabilize the entire system. Hence, the optimum switching states of the rectifier needed to accomplish the desired operation are selected from the Lyapunov energy function derivative so that the stable operation of rectifier is ensured. In doing so, the weighting factor requirement in the cost function is removed. Extensive real time experiments are conducted to reveal the excellent behaviour of the developed control approach. 2 RECTIFIER MODELING AND PROBLEM DEFINITION Rectifier modelling The schematic diagram of a single-phase three-level T-type rectifier supplying dc load bus is presented in Figure 1. It is apparent from Figure 1, there are eight switches whose on and off states can be represented mathematically as follows: S j m = 1 closed 0 open , j = 1 , 2 , 3 , 4 , m = x , y \begin{equation}{S_{jm}} = \left\{ { \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} 1&\quad {{\rm{closed}}}\\[6pt] 0&\quad {{\rm{open}}} \end{array} } \right.,\;\;j = 1,\,2,\,3,\,4,\quad m = x,\,y\end{equation} (1) FIGURE 1Open in figure viewerPowerPoint Schematic diagram of a single-phase three-level T-type rectifier supplying a dc load bus Due to the multi-level ability of the system, the rectifier can produce three distinct pole voltages with respect to the neutral point O. These pole voltages can be produced when the midpoint of each rectifier leg is connected to positive (P), neutral (O) and negative (N) points by means of appropriate switching. The operation states, switching device states and produced pole voltage levels are depicted in Table 1. In the P operation state, the rectifier generates v m O = + V d c / 2 ${v_{mO}} = + {V_{dc}}/2$ when S1m and S2m are closed (ON) and S3m and S4m are open (OFF). In the O operation state, the rectifier produces v m O = 0 V ${v_{mO}} = 0{\rm{ V}}$ when S2m and S3m are ON and S1m and S4m are OFF. Finally, in the N state, the rectifier produces v m O = − V d c / 2 ${v_{mO}} = - {V_{dc}}/2$ when S1m and S2m are OFF and S3m and S4m are ON. For the sake of expressing the multi-level voltage (vxy) and dc side currents, the following control input switching functions are defined: S 1 = S 1 x − S 1 y \begin{equation}{S_1} = {S_{1x}} - {S_{1y}}\end{equation} (2) S 2 = S 2 x − S 2 y \begin{equation}{S_2} = {S_{2x}} - {S_{2y}}\end{equation} (3) TABLE 1. Operation states, switching states and pole voltages Operation states S1m S2m S3m S4m vmO P 1 1 0 0 +Vdc/2 O 0 1 1 0 0 N 0 0 1 1 −Vdc/2 Analysing Table 1, one can obtain the multi-level voltage and neutral current shown in Figure 1 in terms of S1 and S2 as v x y = S 1 V C 1 + S 2 V C 2 \begin{equation}{v_{xy}} = {S_1}{V_{C1}} + {S_2}{V_{C2}}\end{equation} (4) I n = I C 2 − I C 1 = ( S 2 − S 1 ) i s \begin{equation}{I_n} = {I_{C2}} - {I_{C1}} = ({S_2} - {S_1}){i_s}\end{equation} (5) Substituting switching state values into Equation (4), one can obtain five distinct voltage levels 0, ± V d c / 2 $ \pm {V_{dc}}/2$ and ± V d c $ \pm {V_{dc}}$ . The differential equations of the rectifier feeding a resistive load and a CPL can be written as follows: d i s d t = 1 L ( e s − v x y − r i s ) \begin{equation}\frac{{d{i_s}}}{{dt}} = \frac{1}{L}({e_s} - {v_{xy}} - r{i_s})\end{equation} (6) d V d c d t = 2 C 1 I o 1 − V d c R − P CPL V d c \begin{equation}\frac{{d{V_{dc}}}}{{dt}} = \frac{2}{{{C_1}}}\left( {{I_{o1}} - \frac{{{V_{dc}}}}{R} - \frac{{{P_{{\mathop{\rm CPL}\nolimits} }}}}{{{V_{dc}}}}} \right)\end{equation} (7)where v x y = u V d c ${v_{xy}} = u{V_{dc}}$ , I o 1 = u i g ${I_{o1}} = u{i_g}$ , u is the switching function and P CPL ${P_{{\rm{CPL}}}}$ is the power of the CPL. In the derivation of (7), it is assumed that V C 1 = V d c / 2 ${V_{C1}} = {V_{dc}}/2$ and I CPL = P CPL / V d c ${I_{{\rm{CPL}}}} = {P_{{\rm{CPL}}}}/{V_{dc}}$ . The reason of defining CPL as a voltage controlled current source is due to the fact that CPL possesses negative incremental resistance described as R CPL = d V d c d I C P L = d ( P CPL I CPL ) d I CPL = − P CPL I CPL 2 = − R inc \begin{equation}{R_{{\rm{CPL}}}} = \frac{{d{V_{dc}}}}{{d{I_{CPL}}}} = \frac{{d(\frac{{{P_{{\rm{CPL}}}}}}{{{I_{{\rm{CPL}}}}}})}}{{d{I_{{\rm{CPL}}}}}} = - \frac{{{P_{{\rm{CPL}}}}}}{{I_{{\rm{CPL}}}^2}} = - {R_{{\rm{inc}}}}\end{equation} (8) Similarly, IC1 and IC2 can be written in terms of the grid current ( i s ${i_s}$ ), S1 and S2, as follows: I C 1 = 1 2 ( S 1 i s − S 2 i s ) \begin{equation}{I_{C1}} = \frac{1}{2}({S_1}{i_s} - {S_2}{i_s})\end{equation} (9) I C 2 = 1 2 ( S 2 i s − S 1 i s ) \begin{equation}{I_{C2}} = \frac{1}{2}({S_2}{i_s} - {S_1}{i_s})\end{equation} (10) Problem definition As determined in Equation (8), the incremental resistance of CPL is negative which endangers the stability of the rectifier. Now, consider small perturbations in the following variables: e s = e s o + e ∼ s , u = u o + u ∼ i s = i s o + i ∼ s , V d c = V d c o + V ∼ d c \begin{equation} \def\eqcellsep{&}\begin{array}{l} {e_s} = {e_{so}} + {{\tilde e}_s},\quad u = {u_o} + \tilde u\\[6pt] {i_s} = {i_{so}} + {{\tilde i}_s},\quad {V_{dc}} = {V_{dco}} + {{\tilde V}_{dc}} \end{array} \end{equation} (11)where e s o ${e_{so}}$ , u o ${u_o}$ , i s o ${i_{so}}$ and V d c o ${V_{dco}}$ represent the steady-state values of e s ${e_s}$ , u $u$ , i s ${i_s}$ and V d c ${V_{dc}}$ , respectively. Substituting Equation (11) into Equations (6) and (7), one can obtain the following small-signal equations: d i ∼ s d t = 1 L ( e ∼ s − r i ∼ s − u o V ∼ d c − V d c o u ∼ ) \begin{equation}\frac{{d{{\tilde i}_s}}}{{dt}} = \frac{1}{L}({\tilde e_s} - r{\tilde i_s} - {u_o}{\tilde V_{dc}} - {V_{dco}}\tilde u)\end{equation} (12) d V ∼ d c d t = 2 C 1 u o i ∼ s + i s o u ∼ − ( P C P L R + 2 V d c o V ∼ d c ) R V d c o \begin{equation}\frac{{d{{\tilde V}_{dc}}}}{{dt}} = \frac{2}{{{C_1}}}\left( {{u_o}{{\tilde i}_s} + {i_{so}}\tilde u - \frac{{({P_{CPL}}R + 2{V_{dco}}{{\tilde V}_{dc}})}}{{R{V_{dco}}}}} \right)\end{equation} (13) The derivative of Equation (13) can be written as d 2 V ∼ d c d t 2 = 2 C 1 u o d i ∼ s d t + i s o d u ∼ d t + u ∼ d i s o d t − 2 R d V ∼ d c d t \begin{equation}\frac{{{d^2}{{\tilde V}_{dc}}}}{{d{t^2}}} = \frac{2}{{{C_1}}}\left( {{u_o}\frac{{d{{\tilde i}_s}}}{{dt}} + {i_{so}}\frac{{d\tilde u}}{{dt}} + \tilde u\frac{{d{i_{so}}}}{{dt}} - \frac{2}{R}\frac{{d{{\tilde V}_{dc}}}}{{dt}}} \right)\end{equation} (14) Now, substituting Equation (12) into Equation (14) yields d 2 V ∼ d c d t 2 = 2 C 1 u o L ( e ∼ s − r i ∼ s − u o V ∼ d c − V d c o u ∼ ) + i s o d u ∼ d t + u ∼ d i s o d t − 2 R d V ∼ d c d t \begin{eqnarray} \dfrac{{{d^2}{{\tilde V}_{dc}}}}{{d{t^2}}} &=& \dfrac{2}{{{C_1}}}\left(\dfrac{{{u_o}}}{L}({{\tilde e}_s} - r{{\tilde i}_s} - {u_o}{{\tilde V}_{dc}} - {V_{dco}}\tilde u) + {i_{so}}\dfrac{{d\tilde u}}{{dt}}\right.\nonumber\\ && +\, \left.\tilde u\dfrac{{d{i_{so}}}}{{dt}} - \dfrac{2}{R}\dfrac{{d{{\tilde V}_{dc}}}}{{dt}}\right) \end{eqnarray} (15) The transfer functions from ac grid voltage to dc load voltage can be obtained as H 1 ( s ) = V ∼ d c ( s ) e ∼ s ( s ) = 2 u o L C 1 ( s 2 + 4 R C 1 s + 2 u o 2 L C 1 ) \begin{equation}{H_1}(s) = \frac{{{{\tilde V}_{dc}}(s)}}{{{{\tilde e}_s}(s)}} = \frac{{2{u_o}}}{{L{C_1}({s^2} + \frac{4}{{R{C_1}}}s + \frac{{2u_o^2}}{{L{C_1}}})}}\end{equation} (16) H 2 ( s ) = V ∼ d c ( s ) e ∼ s ( s ) = 2 u o L C 1 s 2 + 2 u o 2 \begin{equation}{H_2}(s) = \frac{{{{\tilde V}_{dc}}(s)}}{{{{\tilde e}_s}(s)}} = \frac{{2{u_o}}}{{L{C_1}{s^2} + 2u_o^2}}\end{equation} (17) While H 1 ( s ) ${H_1}(s)$ is derived when a resistive load (R) and a CPL are connected to the load bus, H 2 ( s ) ${H_2}(s)$ is derived when the load bus contains a CPL only. It is obvious that the system is stable while the rectifier supplies a resistive load together with a CPL. However, the system becomes unstable when the rectifier feeds a CPL only. According to the Routh–Hurwitz stability criterion, this instability happens due to the missing s term in the denominator of H 2 ( s ) ${H_2}(s)$ . Therefore, a control method which would guarantee the stability under all operating conditions is needed. 3 FCS-MPC BASED ON LYAPUNOV'S STABILITY THEORY Derivation of Lyapunov energy function The essential control objectives set for the rectifiers include regulation of dc voltage (load voltage), unity power factor operation, reasonably small total harmonic distortion (THD) in the grid currents, fast transient response to sudden load variations and stabilized operation with different loads. It should be mentioned that the control of capacitor voltages (VC1 and VC2) can be accomplished when the control of load voltage ( V d c ${V_{dc}}$ ) is secured. In addition, for the sake of achieving unity power factor operation, the grid voltage and grid current reference must have zero phase difference: e s = E s s i n ( ω t ) , i g ∗ = I s ∗ s i n ( ω t ) \begin{equation}{e_s} = {E_s}sin(\omega t),\;i_g^* = I_s^*sin(\omega t)\end{equation} (18)where I s ∗ $I_s^*$ is the grid current reference amplitude produced by a proportional-integral (PI) controller by using the load voltage error as follows: I s ∗ = K p ( V d c ∗ − V d c ) + K i ∫ ( V d c ∗ − V d c ) d t \begin{equation}I_s^* = {K_p}(V_{dc}^* - {V_{dc}}) + {K_i}\int {(V_{dc}^* - {V_{dc}})dt} \end{equation} (19)where V d c ∗ $V_{dc}^*$ is the reference of V d c ${V_{dc}}$ , K p ${K_p}$ and K i ${K_i}$ denote the proportional and integral gains, respectively. By making use of Equation (6), one can write the derivative of i s ∗ $i_s^*$ as d i s ∗ d t = 1 L ( e s − v x y ∗ − r i s ∗ ) \begin{equation}\frac{{di_s^*}}{{dt}} = \frac{1}{L}({e_s} - v_{xy}^* - ri_s^*)\end{equation} (20) Unlike the conventional MPC methods that employ a cost function, the developed MPC method uses a Lyapunov energy function. As it is clear from Figure 1, the passive components which store energy are C1, C2 and L. Since capacitors can store the energy rather than dissipating it, the continuous-time Lyapunov energy function should be formulated in terms of x 1 ${x_1}$ and x 2 ${x_2}$ as follows: V ( x ) = 1 2 β 1 x 1 2 + 1 2 β 2 x 2 2 \begin{equation}V(x) = \frac{1}{2}{\beta _1}x_1^2 + \frac{1}{2}{\beta _2}x_2^2\end{equation} (21)where β 1 > 0 ${\beta _1} > 0$ and β 2 > 0 ${\beta _2} > 0$ are the positive constants and x 1 ${x_1}$ and x 2 ${x_2}$ are the error variables which are described as follows: x 1 = V C 1 − V C 2 , x 2 = i s − i s ∗ \begin{equation}{x_1} = {V_{C1}} - {V_{C2}},\quad {x_2} = {i_s} - i_s^*\end{equation} (22) Clearly, x2 in Equation (22) includes grid current reference while x1 is defined as the error between VC1 and VC2. While the first term in Equation (21) is the energy stored in the dc capacitors, the second term is the energy stored in the inductor. Hence, that is why the Lyapunov function is also referred to as the energy function in the literature. The rectifier reaches to its equilibrium point when the error variables are zero ( x 1 = 0 ${x_1} = 0$ and x 2 = 0 ${x_2} = 0$ ). This implies that the total energy decays to zero. In order to ensure the asymptotic stability of the rectifier, the energy should be dissipated. In this study, the control law is designed based on the Lyapunov stability theory introduced in Slotine and Li [31]. The control objectives in dc side are to control load voltage (Vdc) and capacitor voltages (VC1 and VC2). While the first objective is accomplished by the PI controller in Equation (19), the latter can be achieved if x1 is forced to zero. The only possibility to have x1 = 0 is when VC1 = VC2. Therefore, the capacitor voltages are self-balanced without using their references. Taking derivative of x 1 ${x_1}$ and x 2 ${x_2}$ using Equations (6), (9), (10) and (20), one can obtain x ̇ 1 = 1 2 C 1 S 1 i s − S 2 i s − 1 2 C 2 S 2 i s − S 1 i s \begin{equation}{\dot x_1} = \frac{1}{{2{C_1}}}\left( {{S_1}{i_s} - {S_2}{i_s}} \right) - \frac{1}{{2{C_2}}}\left( {{S_2}{i_s} - {S_1}{i_s}} \right)\end{equation} (23) x ̇ 2 = 1 L ( v x y ∗ − v x y − r x 2 ) \begin{equation}{\dot x_2} = \frac{1}{L}(v_{xy}^* - {v_{xy}} - r{x_2})\end{equation} (24) Lyapunov energy function has to hold the following features [32, 33]: V ( x ) > 0 $V(x) > 0$ for x 1 ≠ 0 and x 2 ≠ 0 ${x_1} \ne 0\;\,{\rm{and}}\;{x_2} \ne 0$ V ( x ) → ∞ $V(x) \to \infty $ for ∥ x 1 ∥ → ∞ and ∥ x 2 ∥ → ∞ $\| {{x_1}} \| \to \infty \;\,{\rm{and}}\,\;\| {{x_2}} \| \to \infty $ V ̇ ( x ) < 0 $\dot V(x) < 0$ According to the Lyapunov stability theory [31], these features that capture the idea of moving high energy to low energy should be considered to achieve asymptotic stability. Obviously, while the first two features hold, the third feature ( V ̇ ( x ) < 0 $\dot V(x) < 0$ ) that ensures the stability of the rectifier should also be met [34]. Since the Lyapunov function ( V ( x ) $V(x)$ ) in Equation (21) is an energy function, the main goal is to force x 1 ${x_1}$ and x 2 ${x_2}$ to zero as shown in Figure 2. Thus, when x 1 = 0 ${x_1} = 0$ and x 2 = 0 ${x_2} = 0$ converge to zero, V ( x ) $V(x)$ becomes zero as well. The zero-energy is the lowest energy value in the phase-plane ( x 1 − x 2 ${x_1} - {x_2}$ plane). For this reason, the main aim is to minimize V ( x ) $V(x)$ as well as guarantee the negative definiteness of its derivative ( V ̇ ( x ) < 0 $\dot V(x) < 0$ ). FIGURE 2Open in figure viewerPowerPoint The behavior of energy function versus error variables Taking derivative of V ( x ) $V(x)$ yields V ̇ ( x ) = β 1 x ̇ 1 x 1 + β 2 x ̇ 2 x 2 \begin{equation}\dot V(x) = {\beta _1}{\dot x_1}{x_1} + {\beta _2}{\dot x_2}{x_2}\end{equation} (25) Substitution of Equations (4), (23) and (24) into Equation (25) results in V ̇ ( x ) = β 1 2 C 1 ( S 1 i s − S 2 i s ) x 1 − β 1 2 C 2 ( S 2 i s − S 1 i s ) x 1 + β 2 L ( v x y ∗ − S 1 V C 1 − S 2 V C 2 − r x 2 ) x 2 \begin{eqnarray} \dot V(x) &=& \dfrac{{{\beta _1}}}{{2{C_1}}}({S_1}{i_s} - {S_2}{i_s}){x_1} - \dfrac{{{\beta _1}}}{{2{C_2}}}({S_2}{i_s} - {S_1}{i_s}){x_1}\nonumber\\ && +\, \dfrac{{{\beta _2}}}{L}(v_{xy}^* - {S_1}{V_{C1}} - {S_2}{V_{C2}} - r{x_2}){x_2} \end{eqnarray} (26) Substituting i s = x 2 + i s ∗ ${i_s} = {x_2} + i_s^*$ into Equation (26) and assuming that C 1 = C 2 = C ${C_1} = {C_2} = C$ , one can obtain V ̇ ( x ) = S 1 x 1 x 2 β 1 C − β 2 L + S 2 x 1 x 2 β 2 L − β 1 C + β 1 C S 1 − S 2 i s ∗ x 1 + β 2 L v x y ∗ x 2 − S 1 V C 1 x 2 − S 2 V C 2 x 2 − r x 2 2 {\fontsize{8.7}{10.7}{\selectfont{ \begin{eqnarray} \dot V(x) &=& {S_1}{x_1}{x_2}\left( {\dfrac{{{\beta _1}}}{C} - \dfrac{{{\beta _2}}}{L}} \right) + {S_2}{x_1}{x_2}\left( {\dfrac{{{\beta _2}}}{L} - \dfrac{{{\beta _1}}}{C}} \right)\nonumber\\ && +\, \dfrac{{{\beta _1}}}{C}\left( {{S_1} - {S_2}} \right)i_s^*{x_1} +\dfrac{{{\beta _2}}}{L}\left( {v_{xy}^*{x_2} - {S_1}{V_{C1}}{x_2} - {S_2}{V_{C2}}{x_2} - rx_2^2} \right) \nonumber\\ \end{eqnarray}}}} (27) Equation (27) can be simplified further if β 1 ${\beta _1}$ is selected as β 1 = C β 2 / L ${\beta _1} = C{\beta _2}/L$ . In this case, the terms S 1 x 1 x 2 ${S_1}{x_1}{x_2}$ and S 2 x 1 x 2 ${S_2}{x_1}{x_2}$ are eliminated resulting in V ̇ ( x ) = β 2 L ( S 1 − S 2 ) i s ∗ x 1 + v x y ∗ x 2 − S 1 V C 1 x 2 − S 2 V C 2 x 2 − r x 2 2 \begin{eqnarray} \dot V(x) = \frac{{{\beta _2}}}{L}\left( {({S_1}\! -\! {S_2})i_s^*{x_1}\! +\! v_{xy}^*{x_2}\! -\! {S_1}{V_{C1}}{x_2}\! -\! {S_2}{V_{C2}}{x_2} - rx_2^2} \right)\hskip-4pt\nonumber\\ \end{eqnarray} (28) Hence, the stability of rectifier can be assured if V ̇ ( x ) < 0 $\dot V(x) < 0$ . It is worthy to remark that Equation (28) is in continuous-time. In the next subsection, the derivation of controller in discrete-time is presented. Discrete-time model In order to be able to design the FCS-MPC in discrete-time, the derivative of Lyapunov energy function in (28) should be written in discrete-time. Hence, one can define V ̇ x ( n ) ( k + 1 ) $\dot V_x^{(n)}(k + 1)$ at sampling instant ( k + 1 ) t h ${(k + 1)^{th}}$ as follows: V ̇ x ( n ) ( k + 1 ) = β 2 L ( ( S 1 ( n ) ( k ) − S 2 ( n ) ( k ) ) i s ∗ ( k + 1 ) x 1 ( k + 1 ) + v x y ∗ ( k + 1 ) x 2 ( k + 1 ) − S 1 ( n ) ( k ) V C 1 ( k + 1 ) x 2 ( k + 1 ) − S 2 ( n ) ( k ) V C 2 ( k + 1 ) x 2 ( k + 1 ) − r x 2 ( k + 1 ) 2 ) {\fontsize{9.3}{11.3}{\selectfont{ \begin{eqnarray} \dot V_x^{(n)}(k + 1) &=& \frac{{{\beta _2}}}{L}((S_1^{(n)}(k) - S_2^{(n)}(k))i_s^*(k + 1){x_1}(k + 1)\nonumber\\ && +\, v_{xy}^*(k + 1){x_2}(k + 1) - S_1^{(n)}(k){V_{C1}}(k + 1){x_2}(k + 1)\nonumber\\ && -\, S_2^{(n)}(k){V_{C2}}(k + 1){x_2}(k + 1) - r{x_2}{(k + 1)^2})\nonumber\\ \end{eqnarray}}}} (29)where n = 1 , . . . , 9 $n = 1,\;.\;.\;.\;,\;9$ denotes the index which takes values from 1 to 9 since T-type inverter has nine switching states. The control law can be based on selecting the values of appropriate switching functions ( S 1 ( k ) ${S_1}(k)$ and S 2 ( k ) ${S_2}(k)$ ), which would make V ̇ x ( k + 1 ) ${\dot V_x}(k + 1)$ negative. In this case, the rectifier's stability is not endangered as long as β 2 > 0 ${\beta _2} > 0$ . This means that different positive β 2 ${\beta _2}$ values have no influence on the performance of the controller. Hence, β 2 = 1 ${\beta _2} = 1$ is used in this study. This implies that, contrary to the conventional MPC techniques in which the weighting factor is essential and the performance is subject to the weighting factor value, β 2 ${\beta _2}$ used in the developed FCS-MPC technique does not affect the performance at all. Using the definitions in Equation (22), the error variables in discrete-time are described as x 1 ( k + 1 ) = V C 1 ( k + 1 ) − V C 2 ( k + 1 ) \begin{equation}{x_1}(k + 1) = {V_{C1}}(k + 1) - {V_{C2}}(k + 1)\end{equation} (30) x 2 ( k + 1 ) = i s ( k + 1 ) − i s ∗ ( k + 1 ) \begin{equation}{x_2}(k + 1) = {i_s}(k + 1) - i_s^*(k + 1)\end{equation} (31) The values of i s ( k ) ${i_s}(k)$ , V C 1 ( k ) ${V_{C1}}(k)$ and V C 2 ( k ) ${V_{C2}}(k)$ at ( k + 1 ) t h ${(k + 1)^{th}}$ sampling interval are easily obtained by making use of Euler's forward approximation to Equations (6), (9) and (10) as follows: i s ( k + 1 ) = ( 1 − r L T s ) i s ( k ) + T s L e s ( k ) − v x y ( k ) \begin{equation}{i_s}(k + 1) = (1 - \frac{r}{L}{T_s}){i_s}(k) + \frac{{{T_s}}}{L}\left( {{e_s}(k) - {v_{xy}}(k)} \right)\end{equation} (32) V C 1 ( k + 1 ) = V C 1 ( k ) + T s 2 C 1 S 1 ( k ) i s ( k ) − S 2 ( k ) i s ( k ) \begin{equation}{V_{C1}}(k + 1) = {V_{C1}}(k) + \frac{{{T_s}}}{{2{C_1}}}\left( {{S_1}(k){i_s}(k) - {S_2}(k){i_s}(k)} \right)\end{equation} (33) V C 2 ( k + 1 ) = V C 2 ( k ) + T s 2 C 2 S 2 ( k ) i s ( k ) + S 1 ( k ) i s ( k ) \begin{equation}{V_{C2}}(k + 1) = {V_{C2}}(k) + \frac{{{T_s}}}{{2{C_2}}}\left( {{S_2}(k){i_s}(k) + {S_1}(k){i_s}(k)} \right)\end{equation} (34)where T s ${T_s}$ represents the sampling period and v x y ( k ) ${v_{xy}}(k)$ is defined as v x y ( k ) = S 1 ( k ) V C 1 ( k ) + S 2 ( k ) V C 2 ( k ) \begin{equation}{v_{xy}}(k) = {S_1}(k){V_{C1}}(k) + {S_2}(k){V_{C2}}(k)\end{equation} (35) The prediction of e s ( k + 1 ) ${e_s}(k + 1)$ , i s ∗ ( k
Referência(s)