Performance Comparison of Numerical Methods in a Predictive Controller for an AC–DC Power Converter

Abstract

The use of model-based predictive control in power converters has substantially increased in recent years. This control technique always needs a discrete system model to be implemented. There are several methods to obtain a discrete model; in this paper, all common methods are examined from a practical point of view. Their precision, simplicity, and implementation requirements are analyzed to establish their advantages and disadvantages. From this analysis, it is shown that different discretization methods result in different closed-loop converter performance. A model-based predictive control AC–DC converter is used to show that different discretization procedures result in different total harmonic distortion. For this evaluation, a simulation of a 1 kW three-phase active rectifier was performed in Matlab-Simulink.

1. Introduction

Power converters are necessary in any application that requires regulating, storing, distribution, and changing the characteristics of electric energy. General performance of these converters is greatly determined by their controller. That is why the improvement of control algorithms and their implementation represent an incessant research activity. As a consequence, many different control schemes have been developed for power converters.

Hysteresis control compares the measured variable with a reference and determines the switching state of the semiconductor devices; however, it introduces variable switching frequencies, causing resonance problems [1]. Classical techniques, such as proportional–integral (PI) are widely used, but its inclusion in nonlinear systems increases the complexity of the mathematical models and the controller design [2].

The model-based predictive control (MPC) technique has become popular due to its robustness, implementation simplicity, and capability to be applied to different kind of systems [3]. For example, in addition to power converters, predictive control works for robotic systems [4] and industrial applications [5]. When applied to power converters, this control technique uses a discrete time model to control a system variable that is commonly voltage [6], current [7], or torque [8].

In general, the predictive control algorithm uses a system model to predict the behavior of the variable to be controlled and then select the best system operation such that the variable becomes as close as possible to the required behavior [9]. More precisely, it requires a discrete system model to calculate the one-step-ahead value of a variable for all possible switching conditions of the system. Then, the best switch configuration is selected using a cost function that determines the lowest error between the calculated value and its reference. Finally, the appropriate control signals are sent to the system for the next switching period [10].

Until now, to obtain the required discretized model to implement predictive control, most of the previously published works used the Euler method [11,12,13,14] or considered the method to be used irrelevant. This paper shows that the employed method to obtain such a discretized model is not irrelevant, and that different methods give rise to different performances. To this end, an AC–DC converter is used.

An AC–DC converter was selected due to its critical use in medium- and high-power applications, e.g., in renewable energies, for battery recharge in electric and hybrid vehicles [15,16,17]. Conventional AC–DC topologies first perform the AC–DC conversion to correct the power factor [18], followed by a DC–DC conversion to adjust the desired output voltage, using boost or buck topologies [19].

In particular, a conventional three-phase rectifier topology is presented in Figure 1, which includes power transistors with antiparallel diodes as the main power switches. This rectifier operates with high switching frequency and is also known as an active front-end (AFE) rectifier. It overcomes all the problems and limitations of diode and thyristor rectifiers [20]. Its main features are controlled DC voltage, controlled input currents with sinusoidal waveform (reduced harmonics), operation with a very high power factor, and bidirectional operation.

The mathematical models that describe the system behavior are commonly differential equations that must be discretized to work with sampling periods. It is not clear whether different discretization procedures result in different values for the cost function and, hence, affect the converter performance. Discretization procedures are based on numerical methods for solving differential equations, the most common of which is the Euler approximation due to its simplicity [21].

This paper presents a comparison of some numerical methods such as Euler, Runge–Kutta, and trapezoidal approximation of first, second, and third order to obtain the discrete models applied in the predictive controller for the active rectifier of Figure 1. Some aspects to be evaluated are the exactitude, simplicity of the mathematical procedure, processing time, and effect in the THD on the input currents. Comparison results show that the discretization method employed to obtain the required discrete model to implement the predictive control does affect the overall performance of power converters. The obtained results also show that the first-order trapezoidal approximation is a good tradeoff between the obtained performance and implementation simplicity. The comparison was only performed through simulation because the effects that arise in practical implementation would affect all methods equally.

The remainder of the paper is organized as follows: in Section 2, predictive control is precisely described; in Section 3, common methods for system model discretization are revisited; Section 4 describes the cost function employed to select the best switches configuration; the comparison of methods to obtain the required discrete models for predictive control implementation is carried out in Section 5; simulation results are presented in Section 6; in Section 7, the simulations results are discussed; lastly, in Section 8, some conclusions are given.

2. The Predictive Control Technique

A scheme of the general predictive control is shown in Figure 2, the aim is to control the variable X. The first step is to obtain a mathematical expression to model its behavior, commonly a differential equation of n order determined by the specific system characteristics. According to the state variables representation this model may be expressed as follows [22]:

$$\dot{\mathit{X}}=\mathit{A}\mathit{X}+\mathit{B}\mathit{u},$$

(1)

where A is the state matrix, B is the input matrix, and u is the input data.

A discrete expression may be obtained from Equation (1) to calculate the one-step-ahead variable X_k+1 and evaluate all the system operating modes. For example, in power converters, the possible switching combinations of the semiconductor devices produce diverse operating modes.

The specific parameter to be evaluated is contrasted using the cost function, which selects the operating mode that produces the closest value to the reference X_kref and sends the control signals, SW_k, to the system for the next sampling period.

An important step is the discrete model definition, such that a more accurate model results in less error produced between the calculated value and the reference and, thus, closer tracking to the desired reference behavior.

Figure 3 shows an example of X_k and X_kref signals. During the present sampling time (k), the predicted values X_{k +1} obtained for each switching vector state of the control signals are evaluated; the one that generates the closer value to the reference is selected and applied in the next sampling period. The process is performed each sampling period.

3. Numerical Solutions to Obtain Discrete Model

To obtain the discrete model of Equation (1), different numerical solution methods can be used, such as forward and backward Euler, trapezoidal, Runge–Kutta, and compound trapezoidal methods; these methods are analyzed below.

3.1. Forward Euler Method

This is the simplest method to solve a first-order differential equation. The interval $t$ is divided into subintervals $T$ obtaining a set of discrete points k_n as shown in Figure 4, where interval $T$ is the sampling period. At k exists a value X_k; considering that T is very small, an approximation of X_k+1 can be calculated considering a local approximation equivalent to the local slope of X_k, according to

$$\frac{d\mathit{X}}{dt}=\underset{T\to 0}{\lim }\frac{{\mathit{X}}_{k+1}-{\mathit{X}}_k}{T}.$$

(2)

The value of the next sampling period is determined using a local approximation as follows [23]:

$${\mathit{X}}_{k+1}\cong {\mathit{X}}_k+T\left(\frac{d\mathit{X}}{dt}\right).$$

(3)

From Equation (1), it can be written that

$$\left(\frac{d\mathit{X}}{dt}\right)=\mathit{A}{\mathit{X}}_k+\mathit{B}{\mathit{u}}_k.$$

(4)

Substituting Equation (4) into Equation (3), the approximation to obtain the one-step-ahead value is defined as

$${\mathit{X}}_{k+1}\cong {\mathit{X}}_k+T\left(\mathit{A}{\mathit{X}}_k+\mathit{B}{\mathit{u}}_k\right).$$

(5)

3.2. Backward Euler Method

This method considers the derivation of the model from the $k+1$ sampling period, as shown in Figure 4, and is defined by

$$\left(\frac{d{\mathit{X}}_{k+1}}{dt}\right)=\mathit{A}{\mathit{X}}_{k+1}+\mathit{B}{\mathit{u}}_{k+1}.$$

(6)

Substituting Equation (6) into Equation (3), the following expression is obtained:

$${\mathit{X}}_{k+1}\cong {\mathit{X}}_k+T\left(\mathit{A}{\mathit{X}}_{k+1}+\mathit{B}{\mathit{u}}_{k+1}\right).$$

(7)

The function of the next sampling period is calculated solving for ${\mathit{X}}_{k+1}$.

$${\mathit{X}}_{k+1}\cong \left({\mathit{X}}_k-T\mathit{B}{\mathit{u}}_{k+1}\right){\left(\mathit{I}\mathit{d}-T\mathit{A}\right)}^{-1},$$

(8)

where Id is the identity matrix. Considering that $T$ becomes very small and that the local approximation of Equation (2) uses u_k+1 ≈ u_k, the known value u_k may be used instead u_k+1.

$${\mathit{X}}_{k+1}\cong \left({\mathit{X}}_k-T\mathit{B}{\mathit{u}}_k\right){\left(\mathit{I}\mathit{d}-T\mathit{A}\right)}^{-1}.$$

(9)

3.3. Runge–Kutta Method

The Runge–Kutta method derives an approximate solution for a differential equation using a finite number of terms of a Taylor series. In this paper, a fourth-order Runge–Kutta method is analyzed to ensure a smaller error in the approximation. As shown in Figure 5, slopes C₁, C₂, C₃, and C₄ with weights 1, 2, 2, and 1, respectively, are used to obtain slope C_T that crosses the coordinate (k, X_k), which is determined as follows [24]:

$$C_T=\frac{1}{6}\left(C_1+2C_2+2C_3+C_4\right),$$

(10)

where slopes C₁ to C₄ are calculated according to the following equations:

$$C_1=\frac{d{\mathit{X}}_k}{dt},$$

(11)

$$C_2=\frac{d{\mathit{X}}_k}{dt}+\frac{C_1}{2},$$

(12)

$$C_3=\frac{d{\mathit{X}}_k}{dt}+\frac{C_2}{2},$$

(13)

$$C_4=\frac{d{\mathit{X}}_k}{dt}+C_3.$$

(14)

In this way, the Runge–Kutta discrete model to get X_k+1 is obtained as follows:

$${\mathit{X}}_{k+1}\cong {\mathit{X}}_k+T{C}_T.$$

(15)

3.4. Trapezoidal Method

The trapezoidal method is used to solve an integral equation using an approximation that corresponds to the area under the function defined in the interval k and k + 1. According to Figure 6, a trapezium is conformed with the vertices k + 1, k, and u_k, u_k+1.

The approximation of the integral is defined by the trapezium area as follows [24]:

$${\mathit{X}}_{k+1}-{\mathit{X}}_k={\displaystyle \underset{K}{\overset{K+1}{\int }}\mathit{u}(t)dt}\cong \frac{T}{2}\left({\mathit{u}}_{K+1}-{\mathit{u}}_K\right).$$

(16)

The general discrete model for X_k+1 is

$${\mathit{X}}_{k+1}={\displaystyle \underset{K}{\overset{K+1}{\int }}\mathit{u}(t)dt}\cong \frac{T}{2}\left({\mathit{u}}_{K+1}-{\mathit{u}}_K\right)+{\mathit{X}}_k.$$

(17)

3.5. Compound Trapezoidal Method

In the compound trapezoidal method, an interval [a, b] is divided into n intervals that are approximated by a line forming n number of trapeziums, as shown in Figure 7. The total area is obtained by summing all the individual areas as follows [24]:

$${\mathit{X}}_{k+1}={\displaystyle \underset{a}{\overset{b}{\int }}\mathit{u}(t)}dt\cong \frac{T}{2}\left({\mathit{u}}_{k-(n-1)}+2{\displaystyle \sum _{i=0}^{n-2}{\mathit{u}}_{k-i}}+{\mathit{u}}_{k+1}\right)+{\mathit{X}}_k.$$

(18)

This model of order n considers n − 1 previous data.

The accuracy of this numerical method depends on the number of intervals into which [a, b] is divided; more intervals considered results in a closer approximation.

4. Cost Function

The cost function determines the error between the reference X_(k+1)ref and the predicted value for all the possible operating points, X_k+1. The cost function is expressed in orthogonal coordinates (α, β) [9] and measures the error between the references and the predicted value.

$$g=\left|X_{(k+1)\alpha ref}-X_{(k+1)\alpha }\right|+\left|X_{(k+1)\beta ref}-X_{(k+1)\beta }\right|,$$

(19)

where X_(k+1)α and X_(k+1)β are the real and imaginary parts of the predicted load current vector; for a given voltage vector, this prediction is obtained using load model.

The operating mode that generates the minimal error is selected. The future reference values are unknown; however, according to the linear approximation of Equation (19). it can be assumed that X_(k+1)ref ≈ X_kref, taking into account that the sample period T is very close to zero when a system is working with high frequencies.

$${\mathit{X}}_{(k+1)ref}\cong \frac{d{\mathit{X}}_{kref}}{dt}T+{\mathit{X}}_{kref}.$$

(20)

When this assumption is not valid, the future value can be predicted from the present and previous values with a second-order extrapolation.

$${\mathit{X}}_{(k+1)ref}=3{\mathit{X}}_{(k)ref}-3{\mathit{X}}_{(k-1)ref}+{\mathit{X}}_{(k-2)ref}.$$

(21)

Additionally, an extra term can be added to the cost function to consider two or more parameters to be controlled, such that the cost function may be defined as

$$g={\lambda }_a{g}_a+{\lambda }_b{g}_b,$$

(22)

where λ_a and λ_b are the weighting factors that determine the weight of the cost functions g_a and g_b, respectively; the values of the weight factors determine the priority of the control variable [25]. In this paper, only one variable is controlled; hence, weight factors are not considered.

5. Comparison of Discrete Models to Obtain One-Step-Ahead Input Current Vector

The predictive current control technique used to drive the three-phase active rectifier of Figure 1 is shown in Figure 8. The aim of this system is to obtain a DC voltage V_DC and get sinusoidal input currents with a low THD; therefore, the three-phase line current vector i_s is considered the controlled variable. The algorithm is based on discrete samples of the single-phase output voltage of the matrix converter and the three-phase line current, which are processed in the alpha–beta plane. The inverse PQ theory [26] is used to obtain sinusoidal current reference; the amplitude depends on the magnitude of P.

According to

Table 1. The switching states in active rectifier.

Switching State	S1	S2	S3	S4	S5	S6
SW1	0	0	0	1	1	1
SW2	1	0	0	0	1	1
SW3	1	1	0	0	0	1
SW4	0	1	0	1	0	1
SW5	0	1	1	1	0	0
SW6	0	0	1	1	1	0
SW7	1	0	1	0	1	0
SW8	1	1	1	0	0	0

, eight combinations are possible, where switching states 1 and 8 generate i_o = 0 A.

A dynamic model for the derivative of supply currents can be expressed as follows [9]:

$$\frac{d{\mathit{i}}_s}{dt}=\frac{1}{L_s}\left({\mathit{v}}_s-{\mathit{v}}_c-R_s{\mathit{i}}_s\right),$$

(23)

where L_s is the filter inductor, v_s is the supply voltage vector, R_s is the filter resistance, and v_c is the converter voltage vector. To simplify calculations v_s, v_c, and i_s are considered in the bidimensional α–β plane.

$${\mathit{v}}_s=\frac{2}{3}\left(v_{sa}+{\mathit{e}}^{j2\pi /3}{v}_{sb}+{\mathit{e}}^{j4\pi /3}{v}_{sc}\right).$$

(24)

$${\mathit{i}}_s=\frac{2}{3}\left(i_{sa}+{\mathit{e}}^{j2\pi /3}{i}_{sb}+{\mathit{e}}^{j4\pi /3}{i}_{sc}\right).$$

(25)

A discrete model of Equation (23) is required to define the line current for the next sampling period, ${\mathit{i}}_{s_{k+1}}$, for all the possible switching modes. The converter voltage vector, v_c, changes as a function of the switching state vector SW = [S₁, S₂, S₃] according to

$${\mathit{v}}_c=\frac{2}{3}\left(S_1+{\mathit{e}}^{j2\pi /3}{S}_2+{\mathit{e}}^{j4\pi /3}{S}_3\right)V_{DC},$$

(26)

where S₁, S₂, and S₃ are “0” when the switch is opened and “1” when it is closed; S₄, S₅, and S₆ are complementary, and V_DC is the output voltage.

According to the description given in the previous section, Equation (23) can be approximated with the described discrete models to predict the current vector $i_{s_{k+1}}$.

The following considerations are used to obtain the approximations for Euler, Runge–Kutta, and trapezoidal methods:

$${\mathit{x}}_k={\mathit{i}}_{s_k}=\left(i_{a_k}+{\mathit{e}}^{j2\pi /3}{i}_{b_k}+{\mathit{e}}^{j4\pi /3}{i}_{c_k}\right),$$

(27)

$${\mathit{x}}_{k+1}={\mathit{i}}_{s_{k+1}}=\left(i_{a_{k+1}}+{\mathit{e}}^{j2\pi /3}{i}_{b_{k+1}}+{\mathit{e}}^{j4\pi /3}{i}_{c_{k+1}}\right),$$

(28)

$$A=-\frac{R_{s}T}{L_s},$$

(29)

$$B=\frac{1}{L_s},$$

(30)

$${\mathit{u}}_k={\mathit{v}}_{s_k}-{\mathit{v}}_{c_k}.$$

(31)

5.1. Prediction of the One-Step-Ahead Current Using Different Numerical Methods

The obtained expressions for ${\mathit{i}}_{s_{k+1}}$ using the numerical methods detailed in Section 3 are presented below.

• Forward Euler

$${\mathit{i}}_{s_{k+1}}\cong \frac{T}{L_s}\left({\mathit{v}}_{s_k}-{\mathit{v}}_{c_k}\right)+{\mathit{i}}_{s_k}\left(\frac{L_s-R_{s}T}{L_s}\right),$$

(32)

where, ${\mathit{v}}_{c_k}$ is the variable to be tested for all the operating modes.

2.

Backward Euler

$${\mathit{i}}_{s_{k+1}}\cong \left[{\mathit{i}}_{s_k}+\frac{T}{L_s}\left({\mathit{v}}_{s_{k+1}}-{\mathit{v}}_{c_{k+1}}\right)\right]\left(\frac{L_s}{L_s+R_{s}T}\right).$$

(33)

3.

Runge–Kutta

$$C_1=\frac{d{\mathit{i}}_{s_k}}{dt},$$

(34)

$$C_2=\frac{d{\mathit{i}}_{s_k}}{dt}+\frac{C_1}{2},$$

(35)

$$C_3=\frac{d{\mathit{i}}_{s_k}}{dt}+\frac{C_2}{2},$$

(36)

$$C_4=\frac{d{\mathit{i}}_{s_k}}{dt}+C_3,$$

(37)

$$C_T=\frac{1}{6}\left(C_1+2C_2+2C_3+C_4\right),$$

(38)

$${\mathit{i}}_{s_{k+1}}\cong {\mathit{i}}_{s_k}+T{C}_T.$$

(39)

4.

Trapezoidal first order

For the trapezoidal approximation, the integral of Equation (40) is considered to obtain the input current vector:

$${\mathit{i}}_{s_{k+1}}=\frac{1}{L}{\displaystyle \underset{k}{\overset{k+1}{\int }}{\mathit{v}}_{L}dt}+{\mathit{i}}_{s_k}.$$

(40)

The voltage in the inductor v_L is defined according to the difference of voltage supply and voltage in the converter.

$${\mathit{v}}_L={\mathit{v}}_s-{\mathit{v}}_c.$$

(41)

Substituting Equation (41) into Equation (40) and solving to approximate the solution using the first-order trapezoidal method, the predicted input current vector is obtained as follows:

$${\mathit{i}}_{s_{k+1}}\cong \frac{T}{2L_s}\left({\mathit{v}}_{s_{k+1}}-{\mathit{v}}_{c_{k+1}}+{\mathit{v}}_{s_k}-{\mathit{v}}_{c_k}\right)+{\mathit{i}}_{s_k}.$$

(42)

5.

Trapezoidal second order

Using two intervals, the obtained approximation is

$$i_{s_{k+1}}\cong \frac{T}{2L_s}\left[\left({\mathit{v}}_{s_{k-1}}-v_{c_{k-1}}\right)+2\left(v_{s_k}-v_{c_k}\right)+\left(v_{s_{k+1}}-v_{c_{k+1}}\right)\right]+i_{s_k}.$$

(43)

6.

Trapezoidal third order

Using three intervals, the obtained approximation is

$$i_{s_{k+1}}\cong \frac{T}{2L_s}\left[\left(v_{s_{k-2}}-v_{c_{k-2}}\right)+2\left(v_{s_{k-1}}-v_{c_{k-1}}\right)+2\left(v_{s_k}-v_{c_k}\right)+\left(v_{s_{k+1}}-v_{c_{k+1}}\right)\right]+i_{s_k}.$$

(44)

In Equations (42)–(44), ${\mathit{v}}_{c_{k+1}}$ is the parameter to be evaluated using all the possible switching modes. The voltage supply vector, ${\mathit{v}}_{s_k}$, may be used instead of ${\mathit{v}}_{s_{k+1}}$ because they are almost equal. The second-order trapezoidal method uses data at k − 1, whereas the third-order method additionally uses voltages at k − 2.

5.2. Cost Function Evaluation

The minimal error among the eight predicted currents ${\mathit{i}}_{s_{k+1}}$ and the reference i_kref determines the best operating mode of the converter. This minimal error is obtained using

$$g=\left|i_{(k)\alpha ref}-i_{(k+1)\alpha }\right|+\left|i_{(k)\beta ref}-i_{(k+1)\beta }\right|.$$

(45)

Once the switching configuration that causes the minimal error is determined, the appropriate control signals are sent to the semiconductor devices.

6. Simulation Results

A simulation of the three-phase active rectifier of Figure 8 was performed in Matlab-Simulink using the parameters listed in

Table 2. T Simulation parameters.

Parameter	Value
Sampling period Ts	10 µs
Voltage source	127 Vrms
Filer resistor Rs	0.1 Ω
Input filter inductance Ls	10 mH
Source frequency	60 Hz
DC output voltage VDC	300 V

[27] under ideal conditions to evaluate the impact of the use of the different numerical approximations in the predictive controller model. The obtained line current is shown in Figure 9 for Euler, Runge–Kutta, and trapezoidal methods contrasting the input current i_s with the current reference i_ref in phase a.

For this example, the results from the forward and backward Euler methods are almost the same since the resistance $R_s$ is very small; therefore, the values obtained by solving Equations (32) and (33) are very close, and the difference in the i_sa waveforms is imperceptible. When R_s has a greater weight, the Euler backward method tends to produce a closer tracking to the reference.

The error between i_s and i_ref in phase a is shown in Figure 9b; to measure it, the mean square error (MSE) is calculated (

Table 3. Mean square error between i_s and i_ref in phase a for the evaluation of numerical methods in the predictive controller.

	Euler	Runge–Kutta	Trapezoidal 1st-Order	Trapezoidal 2nd-Order	Trapezoidal 3rd-Order
MSE for Ts = 10 µs	0.129520	0.232941	0.038633	0.051673	0.051729
MSE for Ts = 100 µs	3.152851	1.894599	1.366505	3.246525	2.574761

) using Equation (46), where N corresponds to the number of samples [28]. The Runge–Kutta method produces a higher error in contrast with the Euler and trapezoidal approximations, with the trapezoidal method being the strategy that produces a closer tracking to the reference. Therefore, the trapezoidal method provides a higher exactitude, resulting in a smaller current ripple. This is achieved due to the amount of information that is considered to predict the one-step-ahead current, i.e., the data at k and k − 1, in contrast with Euler and Runge–Kutta methods that only employ the data at k.

$$MSE=\frac{1}{N}{\displaystyle \sum _{j=1}^N{\left[i_{ref}(j)-i_s(j)\right]}^2}.$$

(46)

Figure 10 shows a comparison among the first-, second-, and third-order trapezoidal approximations. According to Table 3, the first-order approximation provides the smallest MSE between the reference current and the measured current in comparison with the second- and third-order techniques, such that the current ripple becomes smaller. These methods use data at k, k − 1, and k − 2; as the order of the trapezoidal approximation increases, more previous information is required, increasing the memory use.

To validate the impact of the use of the numerical approximations in the predictive controller model under different conditions, the sampling period was modified to 100 µs. In Figure 11a, the comparison between the reference current and the line current in phase a obtained from Euler, Runge–Kutta, and first-order trapezoidal methods is shown; Figure 11b shows the error. It can be seen that the error increases with the sampling period, but the use of the trapezoidal approximation allows a better performance.

In Figure 12a, a comparison between the current reference and current in phase a is presented when first-, second-, and third-order trapezoidal approximations are used in the predictive model for a T_s = 100 µs. Table 3 reveals that the MSE increases with T_s, but the first-order trapezoidal approximation allows the minimum MSE.

To verify the predictive control capability to operate under dynamic conditions, a power step from 1.5 kW to 2 kW at 50 ms was applied, as shown in Figure 13; the discrete model obtained with the first-order trapezoidal method given by Equation (42) was used in the predictive controller. The current demand increases from 5.8 A to 7.5 A, and the controller continues tracking the reference current.

The bidirectional capability of the AC–DC converter can be observed in Figure 14, where a power step from 1.5 kW to −1.5 kW was applied at 50 ms; at that instant, the current flow changes the direction in the line currents, and the controller continues tracking the new references.

To validate the performance of the controller under dynamical changes in nominal values, some parameters were modified in one phase to unbalance the system. Figure 15 shows the line currents in phase a obtained with Euler and first-order trapezoidal approximations in the controller for a line resistor variation of ±50%. It can be seen that the control action compensates for the unbalance to continue tracking the reference current, and the THD is not significantly altered.

The line currents in phase a obtained with Euler and first-order trapezoidal approximations for an input voltage variation of ±10% are presented in Figure 16. The control action compensates for the variation with the turning on and turning off times in the switching devices, allowing tracking of the reference current without a significant impact on the THD of the input current.

7. Discussion

Table 4. Number of mathematical operations and cycles required.

Numerical Method	Addition	Multiplication	Division	Cycles Required	Time for the Eight Switching States Evaluation (µs)
Euler	3	1	1	328	32.8
Runge–Kutta	9	5	3	986	98.6
Trapezoidal 1st-order	4	2	1	330	33
Trapezoidal 2nd-order	6	3	1	333	33.3
Trapezoidal 3rd-order	8	4	1	336	33.6

shows the number of mathematical operations solved in each discrete model and the cycles needed for them to be executed on a DSPIC; the processor taken as an example was the 32 bit DSPIC33FJ256MC710, where addition and multiplication take one cycle, whereas division takes 182 cycles. Additionally, the time required to complete the evaluation for the eight switching states is shown.

The discrete model obtained with the Runge–Kutta method requires a higher number of cycles to obtain ${\mathit{i}}_{s_{k+1}}$ compared with the Euler strategy, which is the simplest. The number of cycles in the trapezoidal approximation does not increase considerably upon increasing the order; however, the amount of previous information becomes higher, and this should be considered in the design of the predictive controller. It can be seen that the selection of a first-order trapezoidal approximation is suitable to obtain accurate results with a low computational cost.

A better-approximated solution of the mathematical model is reflected in the THD obtained in the supply currents. Figure 17 contrasts the obtained THD when different numerical approximations are used; it can be seen that the Runge–Kutta and first-order Trapezoidal methods generate a low THD, but the Runge–Kutta method requires a higher computational cost. It can also be observed that the use of second- and third-order trapezoidal methods increases the THD in contrast with the first-order trapezoidal method, because the supply currents are deformed a little more.

Table 5. Comparison of numerical methods.

	Euler	Runge–Kutta	Trapezoidal 1st-Order	Trapezoidal 2nd-Order	Trapezoidal 3rd-Order
Exactitude		✔	✔	✔	✔
Simplicity	✔		✔
Fast processing time	✔		✔	✔	✔
Low THD		✔	✔

summarizes the advantages (✔) and disadvantages of all the numerical methods analyzed to obtain the discrete model used in the predictive control algorithm. According to the results, the optimal numerical method is the first-order trapezoidal approximation; it offers exactitude, fast processing time, and a low THD.

8. Conclusions

According to the obtained results, the performance of the predictive control technique can be improved with the use of different numerical methods in the discretization expression used to predict the future behavior of the control variable.

In this paper, the evaluation of five different numerical methods used to obtain the discrete function was performed for application in the predictive controller of a three-phase active rectifier. The simulation results revealed that the use of different numerical methods impacted the THD of the input currents. For example, the THD could be reduced by using more complex numerical methods such as Runge–Kutta; however, the mathematical procedure and computational cost also increased. Other numerical methods offered more exactitude and a lower THD using previous data, such as the trapezoidal approximation; however, the processing time increased to evaluate all possible combinations due to the data to be saved and included in the model.

All the evaluated parameters must be taken into account in the design and model conditions, depending on the application of the predictive control technique and the specific system requirements. The processing time of the mathematical procedure in each numerical method needs to be evaluated for prototype implementation, especially when a high-frequency operation in the power converter is required.

Table 5 revealed that the first-order trapezoidal approximation is a good tradeoff between obtained performance and implementation simplicity; it also allows a low THD, fast processing time, and exactitude for this example, representing a good choice to improve the predictive controller performance in power converter applications.