Predictive Voltage Control Operating at Fixed Switching Frequency of a Neutral-Point Clamped Converter

Uninterruptible power supply units are system formed by power electronics converters to supply sinusoidal voltages to feed critical loads. In this paper, a fixed switching frequency model predictive control strategy is presented for the control of the output voltage in an LC filter connected to a three-level NPC converter. The control objectives of the system are the tracking of the voltage reference and balance of the voltages of the dc-link capacitors. The mathematical model of the converter and the LC filter is developed and the control strategy is explained. Simulation results obtained in the Matlab/Simulink enviroment are presented to validate the control strategy


I. INTRODUCTION
When regulated sinusoidal voltages are required by a load, the solution is to use inverters with an LC output filter to generate ac output voltages with very low harmonic content [1]. One application of this is as the main inverter of an uninterruptible power supply (UPS) system. UPS units are composed of power electronics converters designed to feed critical linear and nonlinear loads such as medical and industrial equipment [2]. For the operation of the inverter the most common approach is to use linear control because of its well-known design and simple implementation. These linear control strategies are PI-based linear cascaded control loops with coordinate transformation (such as the Clarke transform and the Park transform) and modulation techniques such as carrier-based PWM and space vector modulation [3]. The modulation technique is used to linearize the converter and generate the commutation signals based on a time-average principle. Advantages of linear control strategies are the fixed switching frequency and easiness to extend the method to different converter topologies by changing the modulation technique used, etc. but it has some disadvantages such as the difficulty to include contraints and nonlinearities and the necessity of a modulator resulting in slower dynamics. The development of semiconductor technology has increased the processing capability of digital microprocessors and reduced their price allowing the exploration and implementation of new and more complex control schemes. These techniques such as fuzzy control, robust control, sliding mode control and model preditive control (MPC) are more advanced than standard PID control and thus are denominated as advanced control [4]. Predictive control refers to a wide class of controllers such as deadbeat control, hysteresis-based, trayectory-based and MPC [5]. These control techniques shares the same common characteristic which is the use of the mathematical model of the system to predict the future behaviour of the controlled variables over a prediction horizon N to select the appropiate control action based on an optimization criterion [6]. Advantages of predictive control are: applicability to a variety of systems, nonlinearities can be included in the model avoiding the need of linearizing for a given operating point, constraints can be included in the optimization criterion, the multivariable case can be easily included and the possibility to avoid the cascaded structure of linear control schemes resulting in fast transient response. The disadvantages of predictive control strategies are the high computational burden and the need of very good mathematical models of the system under control [4]. FCS-MPC is a predictive control strategy which use the discrete nature of the converter. In every sampling instant the controlled variables are measured and fed to the discrete-time model of the system to predict their future behavior for all possible switching states of the converter and compute a cost function. The cost function depends on the control objetives and can be the absolute error, quadratic error or time-average error between the reference and measured variables [7]. The switching state who minimizes the cost function is stored and applied in the next sampling instant. The strategy does not need a modulator resulting in a variable switching frequency. A FCS-MPC algorithm with constant switching frequency is preferred because it allows easy filter design in applications such as grid-connected converters and inverters with output LC filter [4,8]. A solution to the switching frequency problem is a FCS-MPC algorithm called Modulated Model Predictive Control (M 2 PC). In M 2 PC, a modulation scheme is included in the cost function minimization by selecting and applying, in every sampling instant, two or more switching states with their corresponding application times [9]. This approach has been applied to many power converter topologies including the NPC converter [8][9][10][11][12][13][14] . In this paper, a M 2 PC strategy is proposed for the output voltage control and dc-link capacitor voltage balance of a NPC converter connected to a LC filter feeding a linear resistive load. In Section II the mathematical model of the converter and load is developed, in section III the M 2 PC strategy is explained and in section IV the control strategy is validated with a Matlab/Simulink simulation.

II. TOPOLOGY AND MATHEMATICAL MODEL OF THE CONVERTER
In Fig. 1 a three-level neutral-point clamped (NPC) converter connected to a resistive load through a LC filter is shown.
The NPC converter is a multilevel inverter which means that the converter transforms a fixed dc voltage to a ac voltage with variable magnitude and frequency. The dc-link is formed by two cascaded dc capacitors who provide a floating neutral point (O). The converter topology consist of three phases (or legs) connected in parallel with four high-power switching devices per phase. The high-power switching devices can be either IGBT or GCT [15]. Two series connected diodes (denominated as clamping diodes) are connected to the node between the upper switching devices (with switching signals S 1x and S 2x , with x ∈ {a, b, c}) and the node between the lower switching devices (with switching signals S 1x and S 2x ). Since the converter operation don't allow all the switching devices to be in the ON state at the same time, the lower switching devices work in a complementary manner with the upper switching devices. The node between the clamping diodes is connected to the floating neutral point of the dc-link.
The switching state, S x , summarise the switching devices state (ON or OFF) of the four switches in one of the three phases. Table I shows the relationship between the switching state, the switching signals of the switching devices in one phase and the inverter terminal voltage, v xN (t). The voltage v xN (t) can be expressed as a function of the switching signals of the upper switching devices and the voltages of the dc-link capacitors as follows: The voltages of the dc-link capacitors can be expressed with the differential equations in (2), where i dc1 (t) and i dc2 (t) are the current through the upper capacitor and the current through the lower capacitor, respectively.
The currents through the dc-link capacitors, i dc1 (t) and i dc2 (t), are a function of the output currents i x (t) and the switching states of the converter and can be determined by the equation presented in [16] as follows:  Fig. 2. The plane is divided in six triangular sectors and each sector is divided in four regions. The following current and voltage space vectors are defined for the LC filter and load variables: where a = e j(2π/3) , v(t) is the space vector of the inverter terminal voltages, i(t) is the space vector of the inductors current in the LC filter, v c (t) is the space vector of the capacitors voltages in the LC and i L (t) is the space vector of the load currents. The differential equations who describe the dynamics of the ac side are: where C f , L f y R f are the filter capacitance, filter inductance and filter resistance, respectively. To implement the control algorithm in a digital platform the continuous-time equations need to be discretized. The discrete-time equations are obtained applying the Euler forward method.
Replacing (7) in (2), (5) and (6): III. FIXED SWITCHING FREQUENCY PREDICTIVE VOLTAGE CONTROL M 2 PC is a combination of the operation principles of SVM and FCS-MPC. In every switching instant T s a switching sequence is applied to the converter, the switching sequence consist in the controlled application of the switching states of the voltage vectors that form the region where v ref is located. The transition between one switching state and the next should follow a criterion. The criteria are: (a) at every transition only one leg of the converter can change his switching state and (b) in one sampling interval only two switches per leg can switch states, one for turn-ON and then for turn-OFF. Following these criteria the NPC space vector diagram is analized finding 36 possible switching sequences. Consider region 1 in sector I which is formed by the vectors v 0 , v 1 and v 2 . These are redundant vectors enabling the possibility to form two switching sequences with them (NNN). The difference between both switching sequences is the use of the dclink. The switching sequence of region 1 in sector I is shown in Fig. 3 and the switching sequence of the optimal vectors is shown in Fig. 4, the vector v 0 is the vector with most redundant states in the sequence. The control objetives are: (a) track of the reference voltage and (b) voltage balance in the dc-link capacitors. The following cost funcion is defined: where λ dc is a weighting factor and j ∈ {0, 1, 2}. The discrete equations of the load are used for the prediction of the state variables with a prediction horizon N = 1. The sequence that minimize the global cost function g is applied: The variables d j are the duty cycles of the voltage vectors and depends on the cost function of every vector in the sequence.
The application time of every vector depends on the duty cycles and the sampling time.
In Fig. 5 the block diagram of M 2 PC is shown.

IV. SIMULATION RESULTS
In this section the simulation results of the proposed control strategy are presented. In Table II     Cost function minimization Switching sequence

A. Results in steady state
In Fig. 6 the reference voltage and the voltage in the capacitors of the LC filter is shown in the upper graph and the current through the LC filter inductors is shown in the lower graph. In steady state, the system is capable of following the reference voltage fulfilling the first control objetive.
In Fig. 7 the voltage in the capacitors of the dc-link is shown in the upper graph and the current to the load are shown in the lower graph. The voltages of the dc-link capacitors are kept at half the voltage applied to the dc-link each and the voltage error between v dc1 and v dc2 oscillates between ± 0.1 [V]. The M 2 PC is capable of balance the dc-link capacitors in steady state.

B. Results in transient state
In Fig. 9 the reference voltages and voltages in the capacitors of the LC filter are shown in the upper graph for a step change in the reference from 100 [V] to 150 [V] and the currents in the LC filter inductors is shown in the lower graph. For a step load in the voltage refence, the system is capable of tracking the reference with a fast transient response fulfilling the tracking objetive.
In Fig. 10 the voltage in the capacitors of the dc-link is shown in the upper graph and the currents to the load are shown in the lower graph for a step change in the voltage reference of 100 [V] to 150 [V]. The voltage error between the capacitors increase from ± 0.1 [V] to ± 0.4 [V] at the step time. This is explained because the weigthing factor of the voltage balance objetive in the cost function is not adjusted to the new conditions of the system. Since the difference is small, the control strategy is capable of fulfill the voltage balance in the dc-link capacitors objetive.

V. CONCLUSIONS
In this paper, a fixed switching frequency model predictive control strategy was developed for the voltage control of an LC filter connected to a three-level NPC converter. The mathematical model of the NPC converter and the LC filter is developed and used for the prediction of the states variables