Van-Qua-Binh Ngo

Van-Hiep Vu

Viet-Tuan Pha
email: phamviettuan@gmail.com

Huu-Nhan Nguyen
email: huunhan.vn@gmail.com

Pedro Rodriguez-Ayerbe

Sorin Olaru
email: sorin.olaru@centralesupelec.fr

Hoang-Thinh Do
email: dohoangthinh@tdtu.edu.vn

Lyapunov-Induced Model Predictive Power Control for Grid-tie Three-level Neutral-Point-Clamped Inverter with Dead-time Compensation

Keywords: Computational burden, Control Lyapunov function, Dead-time compensation, Finite control set model predictive control, Stability conditions, Three-level Neutral-Point-Clamped inverter

In this paper, a new power control strategy based on a model predictive power control for the grid-tie three-level neutral point clamped inverter is presented. A dynamical model based on the orientation of grid voltage is used to predict the performance of control variables required for the system. A cost function that includes the tracking power ability, the neutral-point voltage balancing and switching frequency reduction is used to achieve the optimal switching state. A proposed selection scheme of control input is introduced to comprehend the stability of the closed-loop system and reduce the computational cost. At each sampling time, only candidate switching state inputs that guarantee the stability condition through a control Lyapunov function is evaluated in the cost function of the MPC algorithm. Thus, the execution time is remarkably decreased by 26% in comparison with traditional model predictive control. Moreover, the dead-time effect is compensated by incorporating its influence in the proposed prediction model. Simulation and experimental results compared with the traditional model predictive control are used to confirm the effectiveness of the proposed scheme.

I. INTRODUCTION

Multilevel inverters have been extensively applied to high power electronics due to their benefits in the increment of the VOLUME 4, 2016 capacity and in the improvement of the performance. Among them, three-level neutral point-clamped (3L-NPC) inverters become particularly attractive as an alternative solution thanks to their advantageous features such as low total harmonic distortion (THD) of the output current and commonmode voltage [START_REF] Portillo | Modeling Strategy for Back-to-Back Three-Level Converters Applied to High-Power Wind Turbines[END_REF], [START_REF] Schweizer | Comparative Evaluation of Advanced Three-Phase Three-Level Inverter/Converter Topologies Against Two-Level Systems[END_REF]. Moreover, the 3L-NPC inverters have some advantages compared with the conventional two-level inverters, for example, the half voltage stress on the power switching devices and lower dv/dt in the output voltage. The main issue of the 3L-NPC inverter is the unbalance of the DC-link capacitor voltage can be addressed based on several control strategies [START_REF] Portillo | Modeling Strategy for Back-to-Back Three-Level Converters Applied to High-Power Wind Turbines[END_REF], [START_REF] Choi | New Modulation Strategy to Balance the Neutral-Point Voltage for Three-Level Neutral-Clamped Inverter Systems[END_REF]- [START_REF] Ngo | A Simplified Model Predictive Control for T-Type Inverter with Output LC Filter[END_REF].

Recently, the inverters have been widely used to connect the electrical grid. Grid-tie inverters have an important role in applications such as flexible alternating current transmission systems, and renewable energy systems [START_REF] Do | Maximum Power Point Tracking and Output Power Control on Pressure Coupling Wind Energy Conversion System[END_REF], especially the photovoltaic power system [START_REF] Teymour | Solar PV and Battery Storage Integration using a New Configuration of a Three-Level NPC Inverter With Advanced Control Strategy[END_REF]. For this reason, stable and adequate control is necessary for grid-connected inverters. In the past, many control methods have been proposed for gridtie 3L-NPCs. The most popular one is based on the orientation of the grid voltage [START_REF] Wilamowski | Power Electronics and Motor Drives[END_REF] with two control loops to regulate the grid active and reactive power flow. The component of the grid current and DC voltage are regulated by the inner and outer control loops, respectively. This method allows to control the current components and the active/reactive powers independently of each other. However, the transient response is relatively slow. The high-performance control requires precise parameters for the PI current controllers.

In recent years, direct power control (DPC) scheme [START_REF] Malinowski | Virtual-flux-based direct power control of three-phase PWM rectifiers[END_REF] has been introduced to enhance the performance of grid-tie systems. This approach employs a look-up table (LUT) and the angle of the grid voltage or virtual flux to obtain suitable switching states. Hence, this leads to a simple structure due to no necessity of the current control loop and pulse width modulation. Nonetheless, the large power ripples and switching frequency variation are the disadvantages of LUT. Furthermore, to ensure a satisfactory control performance of LUT-DPC, a high sampling frequency is required. In addition, both linear and hysteresis techniques are unable to incorporate the constraints and special necessities such as maximum current, THD of the load current, and switching frequency. To overcome these problems several techniques have been proposed such as proportional resonant [START_REF] Castilla | Reduction of Current Harmonic Distortion in Three-Phase Grid-Connected Photovoltaic Inverters via Resonant Current Control[END_REF], DPC based on fuzzy logic and pulse width modulation [START_REF] Bouafia | Fuzzy-Logic-Based Switching State Selection for Direct Power Control of Three-Phase PWM Rectifier[END_REF], sliding mode control [START_REF] Sebaaly | Design and Implementation of Space Vector Modulation-Based Sliding Mode Control for Grid-Connected 3L-NPC Inverter[END_REF], predictive current control [START_REF] Ngo | Model predictive control with two-step horizon for three-level neutral-point clamped inverter[END_REF]- [START_REF] Kouro | Model Predictive Control: MPC's Role in the Evolution of Power Electronics[END_REF], and predictive power control [START_REF] Ngo | Model predictive power control based on virtual flux for grid connected threelevel neutral-point clamped inverter[END_REF]. In order to reduce the distortion of the output current, neutral-point voltage, and neutralpoint current [START_REF] Imura | Dead-time compensation in model predictive instantaneous-current control[END_REF], [START_REF] Kuznietsov | Model predictive control of a voltage source inverter with compensation of deadtime effects[END_REF], it is necessary to consider the deadtime effect in the control design.

Nowadays, finite control set model predictive control (FCS-MPC) is recognized as a comprehensive control method for power converters thanks to its potential benefits such as simple concept and straightforward incorporation of nonlinearities, additional static constraint, and management of the computational time delay [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF]- [START_REF] Mohamed | A Neural-Network-Based Model Predictive Control of Three-Phase Inverter With an Output LC Filter[END_REF]. In addition, the predictive control achieves a fast dynamic response and an implementation simplicity. Nevertheless, in the FCS-MPC, the optimal control input is obtained from the evaluation of the cost function for all switching states and results in an increased calculation time, particularly with a large amount of switching state and long prediction horizons. To solve this issue, a simplified FCS-MPC for grid connected power converter is proposed in [START_REF] Xia | A Simplified Finite-Control-Set Model-Predictive Control for Power Converters[END_REF] which decreases the computational time by using the equivalent transformation in the cost function and sector distribution of voltage vector. Another method is proposed in [START_REF] Vazquez | Model Predictive Control for Single-Phase NPC Converters Based on Optimal Switching Sequences[END_REF] which employs the best switching sequence MPC to provide fixed switching frequency and maintain restricted computational cost. In [START_REF] Aguilera | Predictive Control of Power Converters: Designs With Guaranteed Performance[END_REF], [START_REF] Aguilera | On stability and performance of finite control set MPC for power converters[END_REF], the authors presented a practical stability theory to obtain a useful cost function by determining the terminal weighting matrix concerning the linear quadratic regulator solution. Previous work has been limited to the nonlinear system due to the balance of DC-link capacitor voltages of the 3L-NPC inverter. A Lyapunov function derived from the predictive model [START_REF] Akter | Modified Model Predictive Control of a Bidirectional AC-DC Converter Based on Lyapunov Function for Energy Storage Systems[END_REF], [START_REF] Kwak | Finite control set predictive control based on Lyapunov function for three-phase voltage source inverters[END_REF] is introduced for the three-phase two-level voltage source converter to address the stability issue. Their solutions are based on the voltage reference vector in the cost function to reduce the computational time. However, this technique not well suited to a multilevel inverter due to the integration of dead-time in the evaluation of the optimization problem with all switching control inputs. Another approach is presented in [START_REF] Ngo | A Simplified Model Predictive Control for T-Type Inverter with Output LC Filter[END_REF] uses sector distribution and feasible candidate inverter voltage to reduce the execution time and maintain the balanced voltage for DC-bus capacitors. Nevertheless, this method is just employed in 3L-T-type inverter with RL load.

Although several studies have been devoted to computational burden, less attention has been paid to consider the stability of the FCS-MPC method. In this paper, the development of a direct power control strategy based on a prediction model for grid-tie 3L-NPC inverter is presented. The proposed method controls the grid power flow, keeps the neutralpoint voltage balancing and reduces the switching frequency. The main contribution of this paper aims to consider the closed-loop stability of the system in the control design. In order to ensure the stability of the system, only candidate switching inputs that guarantee the stability condition derived from a Lyapunov function are taken into account in the evaluation of the cost function. Therefore, the computational burden of the proposed method is reduced by 26% compared to that of the conventional FCS-MPC, resulting in the possibility of real-time implementation. Furthermore, this study focuses on the system modeling which includes the deadtime and switching-time delays caused by the commutated transition of the power converters. In this case, the output and neutral-point voltages can be compensated by using the proposed predictive dynamic model, leading to the improvement of system performance.

The remainder of the paper is structured as follows: Section II presents the dynamic model of a grid-connected 3L-NPC inverter with taking into account the dead-time effect. Next, section III details the proposed Lyapunov functionbased model predictive control for the system. In section IV,

2 VOLUME 4, 2016 1a S 1a D 2a D 2a S 3a S 1 z a D 2 z a D 1b S 1b D 2b D 2b S 1 z b D 2 z b D 1c S 1c D 2c D 2c S 1 z c D 2 z c D ag i 2 C u 2 C 1 C u 1 C Z dc U 3a D 4a S 4a D 3b S 3b D 4b S 4b D 3c S 3c D 4c S 4c D 2 C i 1 C i z i A B C R f L f ag u FIGURE 1: Configuration of the grid-tie 3L-NPC inverter.
a comparative investigation between the proposed method and conventional FCS-MPC is performed in simulation and experimental results. Lastly, the conclusions are outlined in Section V.

II. PREDICTIVE POWER MODEL OF 3L-NPC GRID-TIE INVERTER A. STRUCTURE

Fig. 1 shows a simplified diagram of the grid-tie 3L-NPC inverter. Three possible switching states "1", "0" and "-1" are used to indicate the operating principle of the 3L-NPC inverter. The switching state "1" or "P" denotes two switches S 1a and S 2a in branch A are turned "ON" and the inverter output voltage u AZ is equal to U dc /2, while two switches S 1a and S 2a are switched "OFF" with "-1" or "N", leading to u AZ = -U dc /2. The switching state "0" or "O" signifies two internal switches S 2a and S 3a are turned "ON". According to the direction of the grid current, the output voltage is fixed to zero throughout the NPC diodes D z1a or D z2a . Thus, there are a total of 27 possible configurations with the 3L-NPC inverter. Table 1 summarizes the conduction states produced by the 3L-NPC inverter.

S 1x S 2x S 3x S 4x u AZ current izx 1 ON ON OFF OFF U dc /2 0 0 OFF ON ON OFF 0 ixg -1 OFF OFF ON ON -U dc /2 0

B. MATHEMATICAL MODEL

The output voltage created by the 3L-NPC inverter is formulated in term of phase voltage as:

u inv = 2 3 u AZ + ku BZ + k 2 u CZ , (1)
with k = e j2π/3 = -1 2 + j √ 3

2 . The phase voltage u xZ is computed based on the DC-bus voltage (U dc ) and conduction state (S x ) as follows [START_REF] Ngo | Model predictive control with two-step horizon for three-level neutral-point clamped inverter[END_REF], [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF]:

u xZ = S x U dc 2 , (2)
inv d u k  g Q  Grid g i dc U Grid current reference * gd i * gq i PLL g u z u ^, , g U   Minimization of cost function g 1,..., 27 n  ( 1) abc S k  dc U _ _q , inv dn inv n u u  Z   1,..., 27 n  * _ ( 1) inv q u k  1 z  ( ) abc S k ( ) abc S k   d g u q  g i d i q i  i  i  FIGURE 2: Reference system frames.
where S x indicates the status of the inverter branch: S x ∈ {-1, 0, 1} with x ∈ {a, b, c}.

The dynamic model of grid side inverter can be described by:

u inv = u g + R f i g + L f di g dt , (3)
where

i g = [i ag i bg i cg ] T , u g = [u ag u bg u cg ] T represent
the grid current and voltage. L f , R f signify the filter inductance and resistance, respectively. By employing the Clarke transformation, the inverter output voltage is given as:

u inv = U dc 6 2S a -S b -S c + j √ 3 S b -S c . (4)
The model of the grid side inverter in the dq frame is taken from the Park transformation of (3) as:

u inv_d = R f i d + L f di d dt + u d -ωL f i q , (5)
u inv_q = R f i q + L f di q dt + u q + ωL f i d ,
where ω = 2πf g is the grid frequency. The grid voltage component can be obtained with the assumption of the grid-voltage orientation as shown in Fig. 2:

u d = Ûm , u q = 0, (6)
where Ûm expresses the estimated grid voltage amplitude derived from the phase-locked loop (PLL). Assuming constant DC-link voltage, the dynamic behavior of neutral-point voltage (Z) is achieved from the grid currents and the conduction states as [START_REF] Ngo | A Simplified Model Predictive Control for T-Type Inverter with Output LC Filter[END_REF]:

du z dt = d(u C1 -u C2 ) dt = - 1 C dc i z (7) = 1 2C dc 2 |S a | -|S b | -|S c | i α + √ 3 2C dc |S b | -|S c | i β ,
where

C 1 = C 2 = C dc is the DC-link capacitance. VOLUME 4, 2016
Therefore, the continuous-time model of the system dynamics in the dq synchronous frame can be expressed based on (5), [START_REF] Do | Maximum Power Point Tracking and Output Power Control on Pressure Coupling Wind Energy Conversion System[END_REF], and [START_REF] Teymour | Solar PV and Battery Storage Integration using a New Configuration of a Three-Level NPC Inverter With Advanced Control Strategy[END_REF]:

di d dt = - R f L f i d + 1 L f (u inv_d -Ûm ) + ωi q , ( 8
)
di q dt = - R f L f i q + 1 L f u inv_q -ωi d , du z dt = 1 2C dc 2| S a | -| S b | -| S c | i α + √ 3 2C dc ( S b |) -| S c | i β ,
where the grid current component in the synchronous dq frame is derived from its value in the stationary reference αβ via the rotational coordinate transformation:

i d = i α cos θ + i β sin θ, (9)
i q = i β cos θ -i α sin θ.
The inverter output voltage in the synchronous frame is given in the same way based on (1) and (3) as:

u inv_d = U dc 6 2S a -S b -S c cos θ + √ 3 S b -S c sin θ , u inv_q = U dc 6 √ 3 S b -S c cos θ -2S a -S b -S c sin θ . (10)
The grid active and reactive powers are computed as follows [START_REF] Malinowski | Virtual-flux-based direct power control of three-phase PWM rectifiers[END_REF]:

P g = 3 2 (u d i d + u q i q ) = 3 2 Ûm i d , (11)
Q g = 3 2 (u q i g -u d i q ) = - 3 2 Ûm i q .
Equation [START_REF] Bouafia | Fuzzy-Logic-Based Switching State Selection for Direct Power Control of Three-Phase PWM Rectifier[END_REF] indicates that we can decouple the grid active and reactive power flow via the components of the current in dq axis.

C. AVERAGE VOLTAGE AND CURRENT COMPENSATIONS OF PREDICTION MODEL

The dead-time is required to avoid the simultaneous transition of two switches of each inverter branch in the real system. The effect of the dead-time is to increase the distortion and the ripple of the inverter voltage and grid current leading to an increment of the THD [START_REF] Imura | Dead-time compensation in model predictive instantaneous-current control[END_REF], [START_REF] Kuznietsov | Model predictive control of a voltage source inverter with compensation of deadtime effects[END_REF]. In [START_REF] Ngo | Model Predictive Direct Power Control of Doubly Fed Induction Generator with Dead-Time Compensation[END_REF], the deadtime impact is considered in the prediction model but only incorporating the dead-time delay. In order to decrease the dead-time effects, our study proposes a new average voltage compensation to improve control performance. The inverter voltage during the commutation of switching transition is established based on the previous and existing switching states and the flow of the grid current. Fig. 4(a) shows the commutated transition from the switching state "0" to "1". Switches S 2a and S 3a are turned "ON" at state "0". The NPC diode D z1a is switched "ON" with the positive grid current (i ag > 0). In the ideal case, two switches S 1a and S 3a are turn "ON" and "OFF" simultaneously. Thus, the inverter output voltage steps from zero to U dc /2. When considering the dead-time and turn-on delays, the real switching signals and the output voltage can be approximated as demonstrated in Fig. 3(a). After the dead-time period, S 1a changes from "OFF" to "ON" with the remaining load current i ag . However, it takes the turn-on time to be completely commutated. Then, the load current is switched from D z1a to S 1a and the output voltage is stepped to U dc /2 after the dead-time and turn-on delays as shown in Fig. 4(a). This means that the actual output voltage is reduced by t d + t on and illustrated as the rectangular pattern in Fig. 3(a). On the other hand, in a "1" to "0" transition, S 1a is switched "OFF" and S 3a is commutated "ON" after the dead-time. S 1a is continuously conducted due to the turn-off time t of f , resulting in the remain of the output voltage (U dc /2). Therefore, the output voltage is changed to zero after the turn-off delay. In such case, the conduction state "1" is stretch out by t of f . In summary, the impact of dead-time and switching time delays on the output voltage for each transition is shown in Figs. 3 and4. Consequently, the actual inverter voltage u xZ ˘real generated during a sampling period T s can be given based on Table 2 as:

u xZ-real (k) =                                T s -t d -t on T s u xZ (k), 5 is assured, t of f T s u xZ (k -1), 6 is assured, T s -t of f T s u xZ (k), 7 is assured, t d + t on T s u xZ (k -1), 8 is assured, u xZ (k) otherwise. ( 12
)
TABLE 2: Commutation conditions of the inverter based on the switching state and the flow of the current Case Condition state

1 (Sx(k -1) = 0) & (Sx(k) = 1) 2 (Sx(k -1) = -1) & (Sx(k) = 0) 3 (Sx(k -1) = 1) & (Sx(k) = 0) 4 (Sx(k -1) = 0) & (Sx(k) = -1) 5 ((sign(ixg) > 0) &1) ∨ ((sign(ixg) < 0) &4) 6 ((sign(ixg) > 0) &3) ∨ ((sign(ixg) < 0) &2) 7 ((sign(ixg) < 0) &1) ∨ ((sign(ixg) > 0) &4) 8 ((sign(ixg) < 0) &3) ∨ ((sign(ixg) > 0) &2)
Moreover, the neutral-point voltage is also influenced by the dead-time due to the neutral-point currents as shown in Fig. 4 [START_REF] Sprenger | Experimental verification of direct dead-time Control and DC-link neutral-point balancing of a three level neutral-point-clamped (3L-NPC) VSC[END_REF]. With the aim to explicate this impact, the transition state from "-1" to "0" with i ag < 0 is considered as an illustration (Fig. 4(d)). In "-1" state, S 3a and S 4a are switched "ON" and the current is conducted by S 3a and S 4a . As S 3a is in the state "ON" during the dead-time and state "0" periods, the neutral-point current (i za = i ag ) is transmitted by S 3a and D z2a . The commutation is transferred after S 4a is fully turned "OFF" with turn-off time t of f . Hence, the neutral-point current i za is conducted within T s -t of f time. Therefore, the neutral-point current of one phase which incorporates the dead-time effect can be expressed as:

i zx =                        t d + t on T s i xg , if case 5 is assured, T s -t of f T s i xg , if case 6 is assured, t of f T s i xg , if case 7 is assured, T s -t d -t on T s i xg , if case 8 is assured, (1 -| S x |)i xg , otherwise, (13)
where i zx represents the neutral-point current of one phase.

The dynamic behavior of neutral-point voltage which com- pensates the dead-time effect is rewritten based on [START_REF] Teymour | Solar PV and Battery Storage Integration using a New Configuration of a Three-Level NPC Inverter With Advanced Control Strategy[END_REF]:

du z-real dt = - 1 C dc i z-real = - 1 C dc (i za + i zb + i zc ) , (14)
with i za , i zb , i zc given by [START_REF] Ngo | Model predictive control with two-step horizon for three-level neutral-point clamped inverter[END_REF].

As a result, we can compensate for the dead-time effect by considering its impact with the modified predictive voltage and current in [START_REF] Sebaaly | Design and Implementation of Space Vector Modulation-Based Sliding Mode Control for Grid-Connected 3L-NPC Inverter[END_REF], [START_REF] Ngo | Model predictive control with two-step horizon for three-level neutral-point clamped inverter[END_REF], and [START_REF] Moon | A Robust Deadbeat Finite Set Model Predictive Current Control Based on Discrete Space Vector Modulation for a Grid-Connected Voltage Source Inverter[END_REF].

III. PROPOSED LYAPUNOV MODEL PREDICTIVE POWER CONTROL

The proposed predictive control strategy aims to:

• Pursue grid active and reactive power references.

• Conserve the capacitor voltage balancing of DC-bus.

VOLUME 4, 2016

• Diminish the switching frequency.

With the purpose of fulfilling these control goals, the cost function of the grid-tie 3L-NPC inverter with a computational time delay can be expressed [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF], [START_REF] Rodriguez | State of the Art of Finite Control Set Model Predictive Control in Power Electronics[END_REF]:

g(u k+1 ) = P * g (k + 2) -P p g (k + 2) + λ uz |u p z (k + 2)| + Q * g (k + 2) -Q p g (k + 2) + λ sw n sw , (15)
where P * g (k + 2), Q * g (k + 2) and P p g (k + 2), Q p g (k + 2) stand for the future reference and predicted values of the active and reactive powers at instant k + 2. λ uz , λ sw indicate the weighting elements of the voltage balancing of DC-bus and reduced switching frequency, respectively.

n sw is considered as the auxiliary constraint of the cost function to limit the range of switching frequency. Hence, it can be formulated as:

n sw = x=a,b,c |S x (k + 1) -S x (k)|. ( 16
)
The discrete-time evaluation of the grid current during a sampling time T s is achieved utilizing the first-order forward Euler's method for (8):

îd (k + 1) = i d (k) 1 - T s R f L f + T s L f u inv_d (k) -Ûm + T s ωi q (k), (17)
îq (k + 1) = i q (k) 1 - T s R f L f + T s L f u inv_q (k) -T s ωi d (k),
where u inv_d (k), u inv_q (k) denote the real output voltage considering dead-time compensation are calculated by using [START_REF] Sebaaly | Design and Implementation of Space Vector Modulation-Based Sliding Mode Control for Grid-Connected 3L-NPC Inverter[END_REF] and Park transformation.

Similarly, the expression of discrete-time for grid current at sampling instant k + 2 is given by:

i p d (k + 2) = îd (k + 1) 1 - T s R f L f + T s L f u inv_d (k + 1) - T s L f Ûm + T s ω îq (k + 1), (18)
i p q (k + 2) = îq (k + 1) 1 - T s R f L f + T s L f u inv_q (k + 1) -T s ω îd (k + 1),
The dynamic behavior of neutral-point voltage is achieved using [START_REF] Moon | A Robust Deadbeat Finite Set Model Predictive Current Control Based on Discrete Space Vector Modulation for a Grid-Connected Voltage Source Inverter[END_REF]. Then its expression is given by:

ûz (k + 1) = u z (k) - T s C dc i za (k) + i zb (k) + i zc (k) , u p z (k + 2) = u p z (k + 1) - T s C dc i p za (k + 1) + i p zb (k + 1) - T s C dc i p zc (k + 1), (19)
where i dc1x (k) and i p dc1x (k+1) are the neutral-point currents estimated based on [START_REF] Ngo | Model predictive control with two-step horizon for three-level neutral-point clamped inverter[END_REF].

In order to reduce the computational cost, the predictive future references can be achieved by simple extrapolation as: The errors of the grid current i gd , i qq are described:

P * g (k + 2) = P * g (k + 1) = P * g (k), (20)
Q * g (k + 2) = Q * g (k + 1) = Q * g (k).
i d = i d -i * d , (21)
i q = i q -i * q ,
where the grid current references are calculated based on ( 11)

i * d = 2P * g 3 Ûm , ( 22
) i * q = - 2Q * g 3 Ûm .
Replacing into the dynamic model given by ( 8), we have the error dynamic system of the grid current as:

d i d dt = - R f L f i d + 1 L f u inv_d + ωi q - di * d dt , (23)
d i q dt = - R f L f i q + 1 L f u inv_q - 1 L f Ûm -ωi d - di * q dt .
Since the sampling frequency of the controller (f s = 20 kHz) is greater than grid frequency (f g = 50 Hz), we can assume that:

di * d dt = 0; di * q dt = 0. (24)
As reported by [START_REF] Bouafia | Fuzzy-Logic-Based Switching State Selection for Direct Power Control of Three-Phase PWM Rectifier[END_REF], the grid current can be used to regulate the power flow. In our study, a control Lyapunov function is employed in the proposed method to guarantee the ability of tracking of their powers P * g , Q * g :

V i d , i q = 1 2 K d i 2 d + 1 2 K q i 2 q , (25)
with derivative

• V i d , i q = K d i d d i d dt + K q i q d i q dt , (26)
where K d and K q are the positive gains. The control input is a combination of switching state S p = [S pa S pb S pc ]

T with x ∈ {a, b, c}, whereas p belongs to a finite set {1, ..., 27}. Furthermore, each control input S px is bounded to a set {-1, 0, 1}. Consequently, the control input S opt is obtained as the result of ( 27):

S opt = arg min u k+1 ∈{-1,0,1} 3 g (u k+1 ) . ( 27
) 6 VOLUME 4, 2016
In a real-time system, a computational time delay is inevitable. To deal with this issue, the classical FCS-MPC [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF], [START_REF] Rodriguez | State of the Art of Finite Control Set Model Predictive Control in Power Electronics[END_REF] employs the approximated values to compensate for the computational requirement. Then, a control input which is obtained by minimizing the cost function at the prediction step k + 2 is implemented to the inverter at the previous instant. As was noted in section II-A, there are 27 switching state inputs of the 3L-NPC inverter. With the traditional FCS-MPC, we have to calculate 27 predictions of the grid current, neutral-point voltage balancing, and grid powers. Moreover, it also requires the measured time of the control variables such as voltage, current to accomplish the control input. Subsequently, a large computational burden is the significant drawback of the conventional FCS-MPC, causing difficulty in the practical implementation of the algorithm at a high sampling rate. To decrease the computational time, we employ a pruning technique based on the decreasing Lyapunov function as shown in Fig. 5. At each sampling time, only switching candidates which have a negative • V are evaluated in the cost function of the MPC algorithm. Figs. 6(a) and 6(b) illustrate the Lyapunov function and its derivative. These figures indicate that the stability of the closed-loop system is ensured when the • V is negative at each moment. As suggested in [START_REF] Prior | Quantized-Input Control Lyapunov Approach for Permanent Magnet Synchronous Motor Drives[END_REF], at least one switching state will ensure the stability at each prediction interval. For constant reference currents at least one of the switches will produce current evolution [START_REF] Yang | Adaptive Reference Model Predictive Control With Improved Performance for Voltage-Source Inverters[END_REF] in order to assure a decreasing of the Lyapunov function [START_REF] Wu | An Optimal Voltage-Level Based Model Predictive Control Approach for Four-Level T-Type Nested Neutral Point Clamped Converter With Reduced Calculation Burden[END_REF]. In addition, not all switches will produce a decreasing in the Lyapunov function, as the symmetric of the switches producing a decreasing of the Lyapunov will produce an increase on this function. Fig. 6(c) shows the effect of each switching state which satisfies the stability condition ( • V < 0) are considered for the minimization loop. Several unstable switching states can be eliminated for prediction and assessment of the cost function as detailed in Fig. 6(c). Therefore, this paper underlines the reduction of computational burden by incorporating the stability condition. With the purpose of illustrate this process, Fig. 6(d) details the selection of 10 switching states. For instance at time t = 0.1632 s, three switching states (state 2, 3, and 4) meet the stability criteria that indicate the online computational burden is dramatically decreased. Then, the cost function is optimized among these possible alternatives to select the best control input. As a result, the state 2 is chosen as the best control input at this moment. Consequently, the computation required of the proposed method is limited concerning 27 switching configurations of the 3L-NPC inverter. Indeed, the number of loop interactions of the optimization problem for the standard FCS-MPC is 164 in contrast with the proposed method of 116. The detail improvement of the computational cost of the proposed approach with respect to traditional FCS-MPC is exhibited in Table 3. It is interesting to remark that, we have a different selection based on Lyapunov candidate function

•

V and ∆V = V (k + 1) -V (k) because of a discrete set control voltage. Mentioning in Fig. 6(d), at instant t = 0.1632 s, there are three switching states (state 2, 3 and 4) which have • V < 0. On the other hand, only state 1 has ∆V < 0 as shown in Fig. 7. Hence, the derivative of the Lyapunov function • V is better than its approximation ∆V to choose the switching decreasing Lyapunov function. This leads to the following formula:

u opt = arg min u k+1 ∈{-1,0,1} 3 g (u k+1 ) . (28) subject to • V (k + 1) < 0 TABLE 3: Online computational load of two control strate- gies.

Interaction term

Standard Proposed FCS-MPC approach îd (k + 1), îq(k + 1)

1 1 i * d (k), i * q (k) 1 1 u p xZ-real (k + 1), i p zx (k + 1) 27 27
•

V (k + 1) 0 27 i p d (k + 2), i p q (k + 2) 27 12 nsw 27 12 u p z (k + 2) 27 12 P p g (k + 2), Q p g (k + 2) 27 12 g(u k+1 ) 27 12
Total 164 116

Finally, the proposed predictive control strategy is illustrated in Algorithm 1.

Algorithm 1: Algorithm of the proposed Lyapunov model predictive power control

Input: i g (k), u z (k), U dc , P * g (k) and Q * g (k) Output: Switching signals S a , S b , S c
Predict the grid current îd (k + 1) and îq (k + 1) from ( 17) Compute the reference of the grid current i * d (k) and i * q (k) from ( 22) Initialize the optimal cost function g opt and switching state x opt for i = 1 to 27 do

Compute the predictions with dead-time compensation: u p xZ-real (k + 1) and i p zx (k + 1) using ( 12) and ( 13) Evaluate the derivative of Lyapunov candidate function:

• V (k + 1) based on (26) if • V (k + 1) < 0 then
Predict the current: i p d (k + 2) and i p q (k + 2) based on [START_REF] Ngo | Model predictive power control based on virtual flux for grid connected threelevel neutral-point clamped inverter[END_REF] Estimate the values: n sw from ( 16) Predict the neutral-point voltage u p z (k + 2) from [START_REF] Imura | Dead-time compensation in model predictive instantaneous-current control[END_REF] Calculate the power: P p g (k + 2) and Q p g (k + 2) based on ( 11

The derivative of Lyapunov function Calculate the cost function g from ( 28) if g < g opt then g opt = g; x opt = i end if end if end for

IV. SIMULATION AND EXPERIMENTAL RESULTS

A. SIMULATION RESULTS

The control strategy is conducted within the Matlab/Simulink environment under different operating conditions to verify the feasibility of the proposed method. Table 4 shows the parameters of the controller and system. The mean absolute percentage error (MAPE) can be utilized to examine the control performance. Its expression is given by:

M AP E = 1 n n k=1 h * k -h k h * k , (29)
where h k and h * k are the measurement and reference values, respectively.

On the other hand, the following expression proposed in [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF] can be used to evaluate the average switching frequency (f sw ) of control method:

f sw = x=a,b,c n sw1x + n sw2x + n sw3x + n sw4x 12T sim , (30)
8 VOLUME 4, 2016
where n sw1x , n sw2x , n sw3x and n sw4x represent the commutation of each inverter branch in the period (T sim ). In order to validate the control performance, a comparative study between the proposed method and the traditional FCS-MPC [START_REF] Rodriguez | Predictive Control of Power Converters and Electrical Drives[END_REF] is investigated under the different operating conditions and the same parameters. First we compare results considering no dead-time effect in the switches. To obtain the average frequency of switches f sw = 5 kHz, the sampling time of the controllers is set at T s = 50 µs. Various steps of the grid power are applied to demonstrate the validity of the proposed control scheme. The initial values of the active and reactive power references are 4 kW and -1.5 kVar, corresponding to the P F = -0.94. A step reference of active power (P g-ref ) from 4 kW to 9 kW is applied at time t = 0.1 s, then returned to 4 kW at t = 0.25 s. Fig. 8(b) shows the output active power and its reference of the proposed method. The reactive power demand is varied from -1.5 kVar to 1.5 kVar at t = 0.2 s relative to the leading power factor and lagging power factor, respectively as presented in Fig. 8(d). As highlighted in Fig. 8, the proposed method has the same control performance compared with the traditional FCS-MPC. Indeed, the active power of two controllers accomplishes steady-state of the dynamic response from 4 kW to 9 kW in about 1.5 ms. The MAPE of the active and reactive powers of the classical FCS-MPC are 3.33% and 12.17% while their corresponding values are 3.34% and 13.27% for the proposed method.

The steady-state performance of the grid current is performed with the THD for the traditional MPC of 1.96% (Fig. 9(b)) in comparison with the proposed method distortion of 2.0% (Fig. 9(d)). Fig. 9 illustrates the dynamic response and its Fast Fourier Transform (FFT) generated from Powergui toolbox, where the THD of the grid current satisfies the IEEE 519 standards. The obtained results indicate that the power ripples and THD of the grid current for the proposed method augment moderately compared to traditional MPC. This phenomenon is an expected trade-off to be accepted for including stability criteria of the closed-loop system in the design control of the FCS-MPC. Furthermore, the average computational time of the proposed technique is 19 µs in contrast to the conventional method of 29 µs in a 2.3 GHz, I5 5300 CPU. This means that we can increase the sampling frequency to improve control performance thanks to 34% reduction in computational burden. As a result, this control strategy underlines the importance of practical applications for the implementation with a low-cost processor. The quantitative comparisons of two controllers which comprehend the power ripples, THD of the current and computation time are summarized in Table 5. Moreover, the unbalance of the DC-bus capacitor voltage is also discussed in our research. As shown in Fig. 10, the capacitor voltages keep the balance with MAPE of neutralpoint voltage for the conventional method of 0.18% and 0.17% for the proposed method, respectively. With the aim to confirm the suppression of dead-time influence, the harmonic spectrum of current obtained with deadtime effects for the uncompensated and compensated case is also investigated in our research. Fig. 11(a) indicates that the amplitude of the order harmonics for the uncompensated case is large due to the dead-time effect. In fact, comparing with the ideal case (Fig. 9(d)) the THD of the grid current for the uncompensated case is increased to 2.9% from 2.0%. On the other hand, the components of the harmonic spectrum are reduced, especially the 2nd, 3rd, 5th, 8th and 10th compared with not dead-time compensation as depicted in Fig. 11(b). In real system, the compensation can not completely eliminate this effect because of the influence of various ingredients such as propagation delay or device non-linearity. However, the of our research might help to have important implications for analyzing and solving the dead-time impact.

To validate the robustness of the proposed approach with parameter variations, a test with a reduction of 50% in the filter resistance and inductance is carried out in this study. In this case, the active power is changed from 4 kW to 9 kW at instant t = 0.15 s while the reactive power is kept to zero. As can be seen in Fig. 12, the proposed method is able to achieve sinusoidal grid current with the small ripple of the current and powers.

B. EXPERIMENTAL RESULTS

A scaled down prototype, depicted in Fig. 13, was built in the laboratory to confirm the proficiency of the proposed control scheme. A DSP of TMS320F28335 was The DC-bus voltage is kept at 600 V. The line-to-line grid voltage is fixed to 220 V and 60 Hz of frequency. The sampling time greatly influences the control performance of the FCS-MPC scheme. Small sampling time will result in high performance of the system but require a fast control processor to ensure a large number of calculations necessary. This leads to an increase in the cost of the system, causing a challenge in industrial applications. On the other hand, the quality of the system is deteriorated with a large sampling period. For this reason, the sampling time is set to 100µs which is suitable for a clock frequency of TMS320F28335 (150 Mhz) and enhances the benefit of the low computational burden of the proposed method. The steady-state and transient behavior are investigated under the change in the active and reactive powers to validate the capability of the control strategy. In the first test, the active power (P g ) and reactive powers (Q g ) are set to 2 kW and 0 Var, respectively. The experimental results of grid power and current in steady-state are demonstrated in Fig. 14. These results prove that the grid power exchanges of the proposed technique are kept constant and closed to their references. As illustrated in Fig. 14(b), the grid current is approximately a sinusoidal waveform due to the imperfect sine wave of grid voltage. Moreover, the proposed control strategy is also investigated in the lagging power factor condition. In this case, the reference of active and reactive powers was set to 2000 W and 1000 Var, respectively. Fig. 15 indicates that the proposed method achieved a proper power tracking ability under the change in the power factor value. In the second test, a comparison with the conventional FCS-MPC is carried out with a step transition in grid active power from 0 to 2 kW to verify the capability of power A Fluke 43B instrument is used to measure the THD of the grid current. As demonstrated in Fig. 17, the THD of the grid current for the conventional method without compensation is higher than its value of the proposed method. We argue that this value is due to the important dead-time, causing a deterioration of the quality of the current. Consequently, we believe that our method can be useful in compensation of deadtime effect for the power converters. The comparison of the execution time between the conventional and the proposed method is demonstrated in Fig. 18. Based on these experiments the proposed algorithm reduces 26% of execution time required from 88 µs to 65 µs compared to the conventional method. Therefore, our research provides a novel solution for addressing the problem due to the computational burden and dead-time effect. This benefit can apply to the low-cost processor, leading to a reduction in the cost of the system.

V. CONCLUSION

This paper described the advantages of a power control scheme based on model predictive control for a grid-tie 3L-NPC inverter. A Lyapunov function is employed in the control design to guarantee the stability of the closed-loop system and decrease the computational burden required compared to the traditional FCS-MPC method. Moreover, the dead-time compensation is implemented directly into the controller while maintaining a satisfactory power tracking ability. A comparative study indicates that the proposed method obtains competitive control performance in terms of power and lower THD of the current in comparison with the conventional approach. Simulation and laboratory results have proved the feasibility of this advanced control strategy.

FIGURE 3 :

 3 FIGURE 3: The impact of dead-time and switching time delays on output voltage.

  Commutation with negative current (a) Commutation from switching state "0" to "1" with iag > 0 Commutation with negative current (b) Commutation from switching state "0" to "1" with iag < 0 Commutation from switching state "-1" to "0" with iag > 0 Commutation from switching state "-1" to "0" with iag < 0

FIGURE 4 :

 4 FIGURE 4: Commutation process during the dead-time.

FIGURE 5 :

 5 FIGURE 5: The modified control algorithm with decreased candidate input.

FIGURE 6 :

 6 FIGURE 6: The dynamic responses for the Lyapunov function and its derivative.

FIGURE 7 :

 7 FIGURE 7: ∆V of Lyapunov function.

  Reactive power of the proposed method

FIGURE 8 :

 8 FIGURE 8: The dynamic response of active and reactive powers for the conventional and proposed method.

  Hz) = 19.54 A, THD = 1.96 6 (b) Harmonic spectrum of the current for the conventional method Hz) = 19.54 A, THD = 2.0 2 (d) Harmonic spectrum of the current for the proposed method

FIGURE 9 :

 9 FIGURE 9: The transient response and harmonic spectrum of the grid current for the conventional method and proposed method without dead-time effect.

FIGURE 10 :

 10 FIGURE 10: Neutral-point voltage response.

FIGURE 11 :

 11 FIGURE 11: Grid current spectrum with and without deadtime compensation.

FIGURE 12 :

 12 FIGURE 12: Grid power and current responses with parameter variations.

FIGURE 13 :

 13 FIGURE 13: Experimental laboratory prototype.

iag [ 5 A

 5 /div] icg [5 A/div] ibg [5 A/div] t = [4 ms/div] FFT of grid current [1 A/div] t = [2.5 kHz/div](a) Three-phase grid current Pg [2 kW/div] Qg [1 kVar/div] icg [5 A/div] uca [500 V/div] t = [10 ms/div] (b) Grid power, voltage and current

FIGURE 14 :

 14 FIGURE 14: Steady-state results of the proposed method under the condition P g = 2000 W and Q g = 0 Var.

FIGURE 15 :

 15 FIGURE 15: Steady-state results of the proposed method under the condition P g = 2000 W and Q g = 1000 Var.

VOLUME 4 ,

 4 2016 Pg [2 kW/div] Qg [1 kVar/div] icg [5 A/div] uca [500 V/div] t = [10 ms/div] (a) Grid power and current responses of the conventional method FFT of inverter output voltage [250 V/div] t = [2.5 kHz/div] uC1 [250 V/div] uC2 [250 V/div] uAB [1000 V/div] t = [4 ms/div] (b) Inverter and capacitor voltages of the conventional method Pg [2 kW/div] Qg [1 kVar/div] icg [5 A/div] uca [500 V/div] t = [10 ms/div] (c) Grid power and current responses of the proposed method FFT of inverter output voltage [250 V/div] t = [2.5 kHz/div] uC1 [250 V/div] uC2 [250 V/div] uAB [1000 V/div] t = [4 ms/div] (d) Inverter and capacitor voltages of the proposed method

FIGURE 16 :

 16 FIGURE 16: Transient responses under the step change of active power from 0 to 2 kW.

FIGURE 17 :

 17 FIGURE 17: THD results of grid current for two control methods.

  μs t = 20 [μs/div] (b) Proposed method

FIGURE 18 :

 18 FIGURE 18: Execution time of two controllers.

TABLE 1 :

 1 Conduction state of the inverter leg with x ∈ {a, b, c}

	Sx	Status of switches	Inverter voltage Neutral-point

TABLE 4 :

 4 Parameters of the controller and system.

	Parameters	Value	Representation
	U dc	600 [V]	DC-bus voltage
	C	1000 [µF]	DC-bus capacitance
	R f	80 [mΩ]	Filter resistance
	L f	10 [mH]	Filter inductance
	Ts	50 [µs]	Sampling time
	fg	50 [Hz]	Grid frequency
	Ug	380 [V]	Grid three-phase voltage
	λuz	0.5	Weighting factor of voltage balance
	λsw	0.01	Weighting factor of switching
			frequency reduction
	K d , Kq	1, 1	Positive gains of Lyapunov function
	ton, t of f	0.11, 0.24 [µs] Turn-on/off time of IGBT
	t d	2 [µs]	Dead-time of IGBT

TABLE 5 :

 5 Comparison of control performance for two methods.

		Method
	Performance	Conventional Proposed
		FCS-MPC	method
	MAPE of Pg	3.33%	3.34%
	MAPE of Qg	12.17%	13.27%
	THD of iag	1.96%	2.0%
	Minimum computation time	17 µs	8 µs
	Maximum computation time	41 µs	30 µs
	Average computation time	29 µs	19 µs

 VOLUME 4, 2016