Avoiding module capacitance over-dimensioning of Modular Multilevel Converters with constrained and unconstrained modulator-based Model Predictive Control

Within the last 15 years, modular multilevel converters (MMCs) have emerged to one of the standard topologies for high voltage (HV) and medium voltage (MV) applications. With its many output voltage levels, the MMC can achieve low harmonic distortions even without a high switching frequency per semiconductor device. As a result, a grid filter is often not necessary and the switching losses are very low. Therefore, grid filter and cooling system do not aggravate the power density of the converter system [1].

However, because each of the MMC arms (series connection of modules) represents a single phase converter with its DC link capacitor distributed among the modules, the difference between the constant DC power and the fluctuating AC power must be buffered in the module capacitors. As a result, a major part of the total volume of typical MMC designs is caused by the volume of the module capacitors [2, 3]. Therefore, the reduction of the required module capacitance value can lead to a significant reduction of the total MMC volume. In typical MMC design procedures (cf. [2, 3]), a limited control performance is assumed, such that voltage margins are included and/or the capacitor voltage fluctuation is limited to a fixed value. This is pictured in the left part of Fig. 1. However, when operating the MMC with a specific power factor, one can reduce the required capacitance value until the available arm voltage (sum of all module capacitor voltages of the MMC arm) is just lower than its maximum and just higher as the requested arm output voltage [4] as shown in the right-hand side of Fig. 1. Operating an MMC with this minimum capacitance value requires a high performance control system, because the so called energy balancing must be very fast and precise to still allow appropriate transient performance without resulting in too high capacitor voltages and/or saturation effects because the available arm voltage is not high enough to track the grid and DC voltage with the output voltage of the MMC. Apart from fast energy balancing during transients/reference changes, also the AC and DC currents must be controlled and unwanted circulating currents must be avoided in steady state to keep the conduction losses at a minimum. This results in a multi-input multi-output (MIMO) control problem.

Therefore, the well known standard cascaded PI-controller structure (cf. e.g. [5]) results in comparably low performance, when considering transients/reference steps with relatively small module capacitance values. Furthermore, controller tuning is tedious, because there are couplings between many of the cascaded and parallel SISO controllers in the control system. Also, time delays due to sensing, communication and/or computation cannot be properly compensated with cascaded PI-controllers. MIMO control schemes aiming for better dynamic performance and more structured controller design/tuning were proposed in [6, 7]. However, all these control schemes cannot consider the system constraints like maximum output voltage of the individual MMC arms, maximum module voltage or arm/grid/DC currents.

The control of MIMO systems with constraints is a typical application area of Model Predictive Control (MPC). MPC algorithms perform a prediction of the system’s future behaviour and optimise the control input to find a trade-off between tracking the (possibly contradicting) control references and the system constraints.

For MMCs, MPC could be beneficial, as one could decrease cost, volume and weight of the MMC by minimizing the module capacitance value without loosing (transient) performance, because the controller takes the system’s constraints into account. For example, the module voltages might hit their maximum value constraints in the future after a reference step. If this is predicted by the control algorithm, a counteracting circulating current can be generated, such that the AC and DC output currents are affected as little as possible by the constraint and an optimal trade off between meeting the constraints and tracking the reference can be implemented.

There have been many predictive control schemes proposed for the MMC in literature. Most of them consider the switching state of each module as the system input (direct MPC, e.g. [8‐10]). This results in a large integer optimisation problem, such that in most cases only one prediction step is considered to not cause a too heavy computational burden. For large MMCs with high module numbers even a single prediction step can cause major computational problems. Attempts to reduce this computational burden have been proposed in [11, 12]. This helps to realise longer prediction horizons for MMCs with an intermediate number of modules but still results in computational problems for MMCs with many modules and therefore also high switching frequencies.

In [13] precomputed optimised pulse patterns (OPP) are modified with an MPC algorithm (MP³C). However, here the MPC controls only the grid currents, while the circulating currents are controlled with Deadbeat or hysteresis controllers. The energy balancing is controlled with conventional SISO controllers. Therefore, no constraints for the arm currents and available arm voltages can be considered.

Instead of controlling the individual modules’ switching states with the MPC directly, a modulator can be used. The modulator typically uses a PWM to implement a given reference voltage for the individual arms. Modulators for MMCs can be designed such that all capacitor voltages within one arm are balanced around a mean value [1] and the voltage reference is implemented at the output of the arm with only small errors as shown in [14].

With a modulator, the MMC can be modelled with an averaged model. This means that no actual switching states are considered. It is rather assumed that each MMC arm can generate a continuous output voltage. This average modelling with a modulator enables to use continuous control set (CCS) MPC algorithms known from non switched systems. Therefore, the computational burden of the MPC algorithm is independent of the number of modules and can be used in many application areas from medium voltage drive systems with few modules to HVDC converters with hundreds of modules.

In [15] MMC average model equations are linearised around the current operation point and a linear MPC algorithm based on a quadratic program (QP) is used to determine the reference voltages for the individual modulators of the MMC arms. The (linear) QP formulation allows long prediction horizons without a too heavy computational burden, as fast and efficient solving methods are available (e.g. [16‐18]).

As shown in [19], the problem of the method proposed in [15] is the error of the prediction model that results from the linearisation which can lead to performance degradation.

There are other CCS MPC schemes for MMCs in literature, such as e.g. [20] proposing interesting concepts with particularly low computational burden, but to the author’s knowledge, there is no method proposed that includes a full constraint set, such that it can be guaranteed that the MMC is operated within its physical limits.

Therefore, in this paper, a periodic linear MPC (pMPC) scheme is presented that avoids large prediction errors, but includes constraints for any current and voltage of the MMC. The contributions and structure of this paper are listed in the following:In short, this paper represents a refinement of [21] together with [22] and new research results, to provide a more complete and comprehensive picture of the influence of the control method on the MMC module capacitor selection.

When using the proposed pMPC formulation, there are some consequences for the MMC design arising. These consequences mainly arise from the formulation of the input constraints (cf. Fig. 6).

First, as depicted in Fig. 7 for an exemplary MMC dataset, the voltage margin between the discretised requested and available arm voltage changes with the sampling frequency of the pMPC. For \(1/T_{\mathrm{s}}=1\,kHz\) (upper part of Fig. 7), the voltage margin between \(t=8\,ms\) and \(9\,ms\) is zero and the maximum of the discretised available arm voltage trajectory is just equal to the maximum allowed available arm voltage. This means, that the minimum required module capacitance is selected for this particular sampling frequency. For higher sampling frequencies as shown on the lower part of Fig. 7, a voltage margin between the steady state trajectories of the discretised available arm voltage and the requested arm output voltage results. Therefore, the module capacitance could be reduced while achieving a similar control performance as with the lower sampling frequency.

Second, the linearisation of the square root relation between arm energy and available arm voltage given in (19) influences the way that the MPC can picture the available arm voltage to constrain the maximum controller output voltage. As shown in Fig. 5, there is an error that results in a lower value for the available arm voltage than actually present in the MMC module capacitors. The consequences are also depicted in Fig. 7, where the blue line represents the actual available arm voltage trajectory and the yellow and red line represent the MPC’s interpretation after the linearisation: The discretised values are always lower or equal the actual value.

Both, the timely discretisation as well as the square root approximation result in the necessity to over-dimension the module capacitance values.

In the following, a procedure how to determine the minimum required module capacitance value for the proposed MPC formulation is described. As presented in [4], the minimum required module capacitance value \(C\) is based on the minimization of the average arm energy, if the maximum available arm voltage \(k_{\text{max}}\cdot V_{\text{dc}}\) and the number of MMC modules per arm \(N\) are fixed. The average arm energy is given by This means, that \(k_{\text{dc}}\) needs to be minimized.

The expression must be true, where \(v_{\text{1u}}(t)\) is the arm output voltage and \(e_{\text{1u}}(t)\) is the change in energy stored in the module capacitors of arm 1u (cf. Fig. 4). One does now search for the minimum \(k_{\text{dc}}\) that fulfils (34). Note that in contrast to the equations given in [4], \(v_{\text{1u}}(t)\) also depends on the output power of the MMC, because the voltage drop across the arm and grid inductors must be considered as well. Additionally, the semiconductor voltage drop is also included as described in [14]. Therefore, with (27), \(v_{\text{1u}}(t)\) is given by for the upper arm of the first phase, where \(V_{\text{f,sc}}\) is the fixed semiconductor voltage drop (forward voltage) and \(R_{\text{sc}}\) the resistive voltage drop of the arms semiconductors (cf. [14] for details).

As pointed out in the previous section, the increased calculation time and therefore the rather low achievable sampling frequency of the proposed online MPC scheme lead to the necessity to over dimension the module capacitors. If a control scheme with similar energy balancing performance could work without online optimisations, the increased module capacitance can also be used to implement a certain voltage/energy reserve (cf. Fig. 7) that makes the consideration of constraints unnecessary. This massively simplifies the real time implementation without major drawbacks in the (transient) control performance.

Therefore, in the following the periodic piecewise linear quadratic regulator (pPLQR) is introduced and evaluated with simulations and experimental results.

In the following, the control performance of the proposed pMPC and pPLQR controllers are compared with the classical cascaded PI controller as described in [5]. Therefore, all three controllers have been implemented in a Matlab/Simulink simulation using PLECS blockset to model the MMC. To save simulation time and to exclude the PWM non-idealities from the controller comparison, an averaged model of the MMC as proposed and evaluated in [27] has been implemented. Therefore, no current ripple can be observed in the simulation results presented in this section.

In this section, experimental results obtained with a 12 module MMC prototype are presented. The MMC parameters are listed in Table 1.

The PWM was implemented as proposed in [14] in order to eliminate potential disturbances as far as possible. The communication between the Cyclone V SoC based central controller and the modules is implemented using the SyCCo-Bus [29].

During the design procedure of an MMC, usually a control margin is added between the maximum requested arm voltage and the minimum available arm voltage. Furthermore, the rated trajectory of the available arm voltage is designed to have some margin, before it reaches its maximum allowed value (cf. Fig. 1). These margins are implemented with the module capacitance that provides additional energy storage. Reducing or even omitting the margins can save volume, weight and cost in future MMC implementations.

However, the low control margins place high requirements on the MMC’s control scheme, because the arm energies must stay close to their references even during transients, such as reference steps. Otherwise, the maximum allowed available arm voltage and therefore also the maximum module voltage could be exceeded. Furthermore, saturation effects could arise, because the available arm voltage is not high enough to implement the arm output voltage requested by the controller, which would result in a loss of controllability. In this paper, high performance control of MMCs has been identified as an enabler to reduce the required module capacitance value to a minimum. Therefore, MMC designs with a low module capacitance value become feasible.

To ensure the save operation of the MMC within its physical limits despite the very low module capacitance values and therefore control margins, control schemes that consider the system constraints are considered to be the best option. The pMPC scheme introduced in this chapter features such a complete constraint set in a linear quadratic problem (QP) formulation. As typical for MPC schemes, the downside of the pMPC is its real-time implementation. Because a numeric optimisation problem has to be solved in every sampling step, high sampling frequencies beyond 2 kHz are not reasonable to implement. In combination with the specific input constraint formulation, the module capacitors must be over-designed in order to avoid circulating currents in steady state. A procedure how to properly choose the minimum module capacitance value has been introduced. Simulation results show that the pMPC provides excellent transient and steady state control performance. Power inversions in the millisecond range are possible. Therefore, the pMPC is the method of choice for very fast transient performance without heavily over-dimensioning the module capacitors.

Because the proposed pMPC requires some additional module capacitance compared to the absolute minimum value, this additional energy buffer can be used to gain some control margin for controllers that do not consider the system constraints. Therefore, the pPLQR controller is proposed. It is based on the same cost-function and prediction model as the pMPC, but can be run with a much higher sampling frequency, because no online optimisation must be conducted. Simulation and experimental results show that the pPLQR is a good alternative to the pMPC, if power inversions in less than a grid period are fast enough for the target application. Therefore, the pPLQR can be used to operate MMCs with very low module capacitances due to its excellent energy balancing performance. The pPLQR is simple to implement on embedded hardware platforms, such that it can replace traditional cascaded PI controllers.