A New Backward Euler Stabilized Optimum Controller for NPC Back-to-Back Five Level Converters
This paper presents a backward Euler stabilized-based control strategy applied to a neutral point clamped (NPC) back-to-back connected five level converters. A generalized method is used to obtain the back-to-back NPC converter system model. The backward Euler stabilized-based control strategy uses one set of calculations to compute the optimum voltage vector needed to reach the references and to balance the voltage of the DC-bus capacitors. The output voltage vector is selected using a
... cost functional that includes variable tracking errors in the functional weights, whereas in classic approaches, the weights are considered constant. The proposed modified cost functional enables AC current tracking and DC-bus voltage balancing in a wide range of operating conditions. The paper main contributions are: (i) a backward Euler stabilized-based control strategy applied to a double, back-to-back connected, five level NPC converter; (ii) the use of cost functional weight varying as a function of the controlled variable tracking errors to enforce the controlled variables and to balance the DC capacitor voltages; and (iii) the demonstration of system feasibility for this type of converter topology and control strategy, ensuring a high enough computational efficiency and extending the modulation index from 0.6 to 0.93. Experimental results are presented using a prototype of a five level NPC back-to-back converter. Energies 2017, 10, 735 2 of 16 the voltage imbalance of the DC-bus capacitors, which has been an active research topic using external circuits [8, 9] , modifying pulse width modulation (PWM) techniques [10-13], space vector modulation (SVM) [3, 4, 14] , sliding mode control exploiting converter vector redundancies  , and predictive control    . Some of these techniques require a significant computing power, or have limitations when redundant vector-based strategies are used to balance the capacitor voltages. The theoretical maximum output modulation index is around 0.6 for a back-to-back connected NPC converter with an active load and zero active power exchange, when using the SVM-based control strategy [3, 4] . Known NPC modulation strategies such as PWM and SVM [10, 12] while operating at a constant switching frequency, do not guarantee that controlled outputs are free from DC-bus voltage disturbances, semiconductor "ON" voltages, dead times, or switching delays. Hysteretic control methods are robust to semiconductor non-idealities, load changes, and disturbances, and present fast dynamic responses. Their major drawback is the variable switching frequency, which depends on the operating conditions and load parameters. For some quality indexes, hysteretic control methods may need higher switching frequencies when compared to PWM or SVM modulation techniques [17, 18] . Optimum predictive control techniques drive the output errors towards zero by minimizing the cost functional in each sampling period     . Given the controlled output references, the first step of the NPC predictive controller is to sample the state variables. The second step uses a non-linear model of the system to predict values of the state variables in the next sampling intervals for every possible NPC switching configuration (termed the vector). This requires a powerful numerical processor to compute all the possible future values of the state variables in a sampling step well below 100 µs, to allow switching frequencies around 5 kHz. The last step computes the cost functional for all NPC vectors and chooses the vector that gives the minimum cost functional value in that sampling interval. These three steps are repeated in the next sampling time. Predictive controllers for power electronic converters seem to be a potential alternative since they are well suited to control variables (e.g., currents, voltages, power) presenting coupled dynamics, and can offer closed loop dynamics with decoupled behavior  . However, in each sampling time, predictive algorithms must compute the state variable values in the next sampling interval for all of the possible NPC vectors, together with the corresponding cost functional, requiring a powerful processing unity for converters with available vectors in excess of 27 (three level converters). Predictive algorithms used to reduce time consumption have been reported by [23, 24] . These algorithms use the system inverse dynamics to directly compute the necessary output voltage vector required to track references, while predictive controllers estimate the output errors for all the available vectors. The output voltage vector is then selected among those which are available by minimizing a cost functional that computes the distance between the optimal voltage vector and the existing voltage vectors. However, in , the voltage balancing problem was not addressed but only pointed out briefly in cases where the converter presented redundancies. In  , the voltage balancing problem was solved for three level inverters, but the dependence on non-modeled dynamics is not addressed. This paper uses a stable method to compute the necessary output voltage vector and extends the voltage balancing to five level NPC converters, where balancing is more challenging, by using an approach that is valid even if there are no redundant vectors. In  , only constant weights are used in the quadratic cost function, while the proposed paper uses variable weights as a function of variable tracking errors, for the cost functional equations of functions. The approach proposed here, while not needed in three level NPCs, is nearly mandatory in five level inverters, as balancing the four DC capacitor voltages using 250 vectors is, at least, more complex and difficult. The approach of this paper enables the enlargement of NPC voltage balancing range for different active and reactive power flow conditions. In view of these problems, this paper presents a backward Euler stabilized control strategy applied to a back-to-back five level NPC converter to control the line inject AC currents and to balance the four capacitor voltages. The paper starts with the back-to-back converter modeling, using a systematic switching variable generalized for m level converters (Section 2).