3.3 Model Predictive Control for Power Electronics and Drives

Although the theory of MPC was developed in the 1970s, its application in power electronics and drives is more recent due to the fast sampling times that are required in these systems. The fast microcontrollers available in the last decade have triggered research in new control schemes, such as MPC, for power electronics and drives.

As mentioned previously, MPC includes a very wide family of controllers and several different implementations have been proposed. An interesting alternative is the use of generalized predictive control (GPC), which allows solution of the optimization problem analytically, when the system is linear and there are no constraints, providing an explicit control law that can be easily implemented [11, 12]. This control scheme has been used in several power converter [13–15] and drive applications [16–18].

In order to make possible the implementation of MPC in a real system, considering the little time available for calculations due to the fast sampling, it has been proposed to move most of the optimization problem offline using a strategy called explicit MPC. The optimization problem of MPC is solved offline considering the system model, constraints, and objectives, resulting in a look-up table containing the optimal solution as a function of the state of the system. Explicit MPC has been applied for the control of power converters such as DC–DC converters and three-phase inverters [19, 20], and in the control of permanent magnet synchronous motors [21].

Most GPC and explicit MPC schemes approximate the model of the power converter as a linear system by using a modulator. This approximation simplifies the optimization and allows the calculation of an explicit control law, avoiding the need for online optimization. However, this simplification does not take into account the discrete nature of the power converters.

By including the discrete nature of power converters, it is possible to simplify the optimization problem, allowing its online implementation. Considering the finite number of switching states, and the fast microprocessors available today, calculation of the optimal actuation by online evaluation of each switching state is a real possibility. This consideration allows more flexibility and simplicity in the control scheme, as will be explained in subsequent chapters of this book. As the switching states of the power converters allows finite number of possible actuations, this last approach has been called, in some works, finite control set MPC.

3.3.1 Controller Design

In the design stage of finite control set MPC for the control of a power converter, the following steps are identified:

• Modeling of the power converter identifying all possible switching states and its relation to the input or output voltages or currents.
• Defining a cost function that represents the desired behavior of the system.
• Obtaining discrete-time models that allow one to predict the future behavior of the variables to be controlled.

When modeling a converter, the basic element is the power switch, which can be an IGBT, a thyristor, a gate turn-off thyristor (GTO), or others. The simplest model of this power switches considers an ideal switch with only two states: on and off. Therefore, the total number of switching states of a power converter is equal to the number of different combinations of the two switching states of each switch. However, some combinations are not possible, for example, those combinations that short-circuit the DC link.

As a general rule, the number of possible switching states N is

3.5

where x is the number of possible states of each leg of the converter, and y is the number of phases (or legs) of the converter. In this way a three-phase, two-level converter has N = 2³ = 8 possible switching states, a three-phase, three-level converter has N = 3³ = 27 switching states, and a five-phase, two-level converter has N = 2⁵ = 32 switching states. In some multilevel topologies the number of switching states of the converter can be very high, as in a three-phase, nine-level cascaded H-bridge inverter, where the number of switching states is more than 16 million.

Another aspect of the model of the converter is the relation between the switching states and the voltage levels, in the case of single-phase converters, or voltage vectors, in the case of three-phase or multi-phase converters. For current source converters, the possible switching states are related to current vectors instead of voltage vectors. It can be found that, in several cases, two or more switching states generate the same voltage vector. For example, in a three-phase, two-level converter, the eight switching states generate seven different voltage vectors, with two switching states generating the zero vector. In a three-phase, three-level converter there is a major redundancy, with 27 switching states generating 19 different voltage vectors. Figure 3.3 depicts the relation between switching states and voltage vectors for two different converter topologies. In some other topologies, the method of calculating the possible switching states may be different.

Figure 3.3 Voltage vectors generated by different converters. (a) Three-phase, two-level inverter. (b) Three-phase, three-level inverter

Each different application imposes several control requirements on the systems such as current control, torque control, power control, low switching frequency, etc. These requirements can be expressed as a cost function to be minimized. The most basic cost function to be defined is some measure of error between a reference and a predicted variable, for example, load current error, power error, torque error, and others, as will be

shown in the following chapters of this book. However, one of the advantages of the predictive control methods is the possibility to control different types of variables and include restrictions on the cost function. In order to deal with the different units and magnitudes of the controlled variables, each term in the cost function is multiplied by a weighting factor that can be used to adjust the importance of each term.

When building the model for prediction, the controlled variables must be considered in order to get discrete-time models that can be used for the prediction of these variables. It is also important to define which variables are measured and which ones are not measured, because in some cases variables that are required for the predictive model are not measured and some kind of estimate will be needed.

To get a discrete-time model it is necessary to use some discretization methods. For first-order systems it is useful, because it is simple, to approximate the derivatives using the Euler forward method, that is, using

3.6

where T_s is the sampling time. However, when the order of the system is higher, the discrete-time model obtained using the Euler method is not so good because the error introduced by this method for higher order systems is significant. For these higher order systems, an exact discretization must be used.

3.3.2 Implementation

When implemented, the controller must consider the following tasks:

• Predict the behavior of the controlled variables for all possible switching states.
• Evaluate the cost function for each prediction.
• Select the switching state that minimizes the cost function.

Implementation of predictive models and a predictive control strategy may encounter different difficulties depending on the type of platform used. When implemented using a fixed-point processor, special attention must be paid to programming in order to get the best accuracy in the fixed-point representation of the variables. On the other hand, when implemented using a floating-point processor, almost the same programming used for simulations can be used in the laboratory.

Depending on the complexity of the controlled system, the number of calculations can be significant and will limit the minimum sampling time. In the simplest case, predictive current control, the calculation time is small, but in other schemes such as torque and flux control, the calculation time is the parameter which determines the allowed sampling time.

To select the switching state which minimizes the cost function, all possible states are evaluated and the optimal value is stored to be applied next. The number of calculations is directly related to the number of possible switching states. In the case of the three-phase, two-level inverter, to calculate predictions for the eight possible switching states is not a problem, but in the case of multi level and multi-phase systems, a different optimization method must be considered in order to reduce the number of calculations.

3.3.3 General Control Scheme

A general control scheme for MPC applied to power converters and drives is presented in Figure 3.4. The power converter can be from any topology and number of phases, while the generic load shown in the figure can represent an electrical machine, the grid, or any other active or passive load. In this scheme measured variables x(k) are used in the model to calculate predictions x(k + 1) of the controlled variables for each one of the n possible actuations, that is, switching states, voltages, or currents. Then these predictions are evaluated using a cost function which considers the reference values x^*(k) and restrictions, and the optimal actuation S is selected and applied in the converter.

Figure 3.4 General MPC scheme for power converters