Efficient Fixed-Switching Modulated Finite Control Set-Model Predictive Control Based on Artificial Neural Networks

Abstract

The disadvantage of finite control set-model predictive control (FCS-MPC) is that the switching frequency is variable and relies on the sampling time and operating point. This paper describes how to implement a new algorithm to achieve a fixed-switching frequency functionality for the FCS-MPC. The used approach combines the FCS-MPC with the SVPWM, resulting in the calculation of dwell times and the selection of the best two active vectors for the next sample interval. These dwell times have a significant impact on FCS-MPC performance during transient and steady-state conditions, and their values are determined using various mathematical models. To solve the problem of the fixed-switching frequency with lower harmonics distortion compared to the conventional modulated MPC (M${}^2$PC), an ANN-based trained network is proposed to calculate the duty-cycle of the applied vectors and thus the dwell time in the next sampling interval. The ANN network receives the cost functions of the two active vectors and the zero vector from the M${}^2$PC control algorithm and determines the optimal duty-cycle for each vector based on a proper tuning. In this way, three goals are achieved, the first goal is that the algorithm explicitly obtains a fixed-switching frequency, and secondly, the cost is as simple as the conventional M${}^2$PC. Finally, the feature of including objectives and non-linearity is still applicable. The paper’s case study used the two level voltage source inverter (2L-VSI) for uninterruptible power supply (UPS) applications. The results based on MATLAB/Simulink revealed that the ANN-M${}^2$PC has retained all FCS-MPC features in addition to operating at a fixed-switching frequency, while the power quality is significantly enhanced.

1. Introduction

FCS-MPC has several distinguishing characteristics, including quick response, simple concept, ease of constraint inclusion, and nonlinearity [1,2]. However, due to the action of the FCS-MPC, which is delivered straight forward to the converter switches without any modulation stage, it has the disadvantage of variable switching frequency [3]. Several challenges may develop when the converter runs at a variable switching frequency, including circuit resonance, magnetic component and filter complexity, and a lack of power quality for critical applications [4,5].

There are numerous approaches to adjust the frequency and obtain a fixed switching action for converters controlled by the FCS-MPC. The first method is to include an extra term in the cost function that accounts for the converter switches’ previous switching patterns [6]. In addition to the fundamental disadvantage of not being able to choose from a restricted number of acceptable switching states due to the lack of a modulator, this method necessitates an additional weighting component, which could make the tuning of cost function more difficult [7]. Another option is to employ band-stop filters (i.e., notch filters) to regulate the harmonics. This strategy may provide problems with stability, particularly in grid-connected applications [8]. The alternative way is to employ the periodic control technique, which is based on assessing up and down switching events to achieve the modulation behavior [9]. In this procedure, a correctly tuned factor is also required for many operations. Another option is to remove the excess term and use a modulation technique like space vector modulation (SVM) [10]. In this scenario, the FCS-MPC and SVM combination provides a decent performance in terms of fast response and fixed-switching frequency [11]. In this approach, there are numerous ways to compute the dwell times associated with each applied vector. One is based on the traditional SVM, while others use virtual methods to calculate dwell time appropriately [12].

The research community’s most recent tendency is to fix the FCS-MPC switching frequency, which is based on the use of modulated MPC [13]. Approaches such as discrete space vector [14], deadbeat predictive control [15,16], and predictive PWM methods result in intricate expressions for switching time calculations [17,18]. Moreover, they are not intuitive, because introducing other objectives into the cost function are difficult during the control design [19,20]. In such methods, the optimal vector is not applied during the whole sampling period, but it is applied in a part of the sampling interval, while in the remaining part, another vector can be applied.

To address these issues, a novel method has been presented that allows operation at a fixed switching frequency, while keeping the benefits of the FCS-MPC controller [21]. This method uses a linear PI controller to simulate the implementation of SVM by using two active vectors and two zero vectors. The time of each vector should be determined and optimized using the same methodology as the conventional SVPWM to reach a certain goal. Dwell times are what they are called, and this is referred to as MMPC or M${}^2$PC [21].

On the other hand, the use of artificial neural networks (ANNs)-based approaches in the field of power electronics is rapidly growing. The use of neural networks for dynamical system control was first proposed in the early 1990s [22]. Multi-layer perceptions were used for a variety of tasks, including system identification and control law implementation. ANN-based controllers and estimators, in particular, have been widely employed to identify and operate power converters and motor drives [23]. The advantages of ANN-based controllers include: they can generally improve the performance of the system when properly tuned, and they can be created based on data received from a real system or plant in the lack of expert knowledge. They do, however, necessitate a considerable amount of training data. However, as the current study shows, this is not a significant disadvantage because trustworthy modeling techniques may be used to acquire data [24,25].

In much prior research, the ANN has been used to replace the predictive controller, which considers the converter as a black box. The main disadvantage of this strategy is that it is limited in terms of flexibility because of the inability to add further constraints to the objective function once the ANN network has been established [26,27]. This leads to the loss of one of the characteristics of the MPC, which is the potential to deal with several constraints, within a single control law. On the other hand, other studies have employed the ANN to fine-tune several targets in order to obtain reduced THD [28].

The objective of this paper is to apply ANN to optimally define the duty-cycle and thus the dwell time associated with the M${}^2$PC method in an accurate way. The technique can reserve the converter operation at a fixed-switching frequency, while the THD improves greatly, and the M${}^2$PC’s properties are preserved. The conventional M${}^2$PC for the 2L-VSI will be discussed in Section 2, while the proposed ANN-M${}^2$PC will be discussed in Section 3. The simulation results and discussion are given in Section 4, while the conclusion is reported in Section 5.

2. Conventional FCS-MPC for the 3-Phase 2L-VSI Based on SVPWM

Many applications have been proposed for MPC, including electrical drivers, UPS, and grid-connected inverters [6]. FCS-MPC is based on the premise of employing a discrete model of the power converter with its associated filter to anticipate the future behavior of all potential control inputs, and then applying the best one that minimizes a predetermined cost function (CF) at each sampling time ($T_s$). The main concept is to harness the microcontroller’s raw processing capability to incorporate all control loops into a single algorithm that takes into account the converter’s model and the accompanying filter.

The general block diagram of FCS-MPC to control a three-phase voltage source inverter with an $LC$ filter in the output is shown in Figure 1. The FCS-MPC procedure is conducted sequentially in its most basic version, and at the start of each sampling time, it applies a new switching pattern that is determined from the previous step. It then receives new and updated measurements in order to select a new switching pattern. The three-phase 2L-VSI power circuit is considered, with the two switches in each leg operating in a complementary mode to avoid short-circuit across the input source. The switching states are represented by $S_a$, $S_b$, and $S_c$ switching signals, which are specified as follows:

$$S_a=\left\{\begin{array}{c}1 \mathrm{if}{S}_1\mathrm{on} \mathrm{and}{S}_4\mathrm{off}\\ 0 \mathrm{if}{S}_1\mathrm{off} \mathrm{and}{S}_4\mathrm{on}\end{array}\right.$$

(1)

$$S_b=\left\{\begin{array}{c}1 \mathrm{if}{S}_2\mathrm{on} \mathrm{and}{S}_5\mathrm{off}\\ 0 \mathrm{if}{S}_2\mathrm{off} \mathrm{and}{S}_5\mathrm{on}\end{array}\right.$$

(2)

$$S_c=\left\{\begin{array}{c}1 \mathrm{if}{S}_3\mathrm{on} \mathrm{and}{S}_6\mathrm{off}\\ 0 \mathrm{if}{S}_3\mathrm{off} \mathrm{and}{S}_6\mathrm{on}\end{array}\right.$$

(3)

The voltage across filter inductance ($L_{fa}$, $L_{fb}$, $L_{fc}$) can be written in a vectorial form as:

$$L_f\frac{\mathrm{d}{I}_{L_f}}{\mathrm{d}t}=V_i-V_{C_f}$$

(4)

where $L_f$ is the filter inductance, $I_{L_f}$ is the filter current, and $V_{C_f}$ is the filter voltage, and $V_i$ denotes the system’s inverter voltage, which has eight different voltage vectors (i.e., six active vectors and two zero vectors), as illustrated in Figure 2. It is clear that the space vector diagram has six sectors. Mathematically, the equation that defines the dynamic behavior of the output voltage is as follows:

$$C_f\frac{\mathrm{d}{V}_{C_f}}{\mathrm{d}t}=I_{L_f}-I_o$$

(5)

where $I_o$ is the load current, which can be measured or estimated, and $C_f$ is the filter capacitance ($C_{fa}$, $C_{fb}$, $C_{fc}$). In the state space model, these equations can be expressed as:

$$\frac{\mathrm{d}X}{\mathrm{d}t}=AX+B_1{V}_i+B_2{I}_o$$

(6)

where

$$X=\left[\begin{array}{c}{I}_{L_f}\\ V_{c_f}\end{array}\right]$$

(7)

$$A=\left[\begin{array}{cc}-R/L_f& -1/L_f\\ 1/C_f& 0\end{array}\right]$$

(8)

$$B_1=\left[\begin{array}{c}1/L_f\\ 0\end{array}\right]$$

(9)

$$B_2=\left[\begin{array}{c}0\\ -1/C_f\end{array}\right]$$

(10)

A discrete model is obtained from (6) and it can be expressed as follows:

$$x\left(\begin{array}{c}k+1\end{array}\right)=A_{q}x\left(\begin{array}{c}k\end{array}\right)+B_q{V}_i\left(\begin{array}{c}k\end{array}\right)+B_{dq}{I}_o\left(\begin{array}{c}k\end{array}\right)$$

(11)

where

$$A_q=ex{p}^{A{T}_s}$$

(12)

$$B_q={\int }_0^{T_s}ex{p}^{A\tau }{B}_{1}d\tau$$

(13)

$$B_{dq}={\int }_0^{T_s}ex{p}^{A\tau }{B}_{2}d\tau$$

(14)

.

This model is used to predict the filter current in $L_f$ and voltage across $C_f$ for any input voltage and different loading conditions. The evaluation of cost function influences the choice of the optimal output voltage vector to be applied in the next sampling interval. For such a system, the main control target is to fix the inverter output voltage, therefore the CF will be defined as follows:

$$\begin{array}{c}\hfill J={(V_{C_{f_{\alpha }}}^{*}-V_{C_{f_{\alpha }}}(k+1))}^2+{(V_{C_{f_{\beta }}}^{*}-V_{C_{f_{\beta }}}(k+1))}^2\end{array}$$

(15)

where $V_{C_f\alpha }^{*}$ and $V_{C_f\beta }^{*}$ are the real and imaginary parts of the reference voltage, respectively.

In space vector modulation, it is possible to define each available vector in the 2L-VSI in ($\alpha \beta$) plane, as shown in Figure 2. The modulated MPC technique calculates the prediction of the two active vectors that compose each sector at every sampling time. The cost function, defined by (15), is evaluated for each sector. Each prediction is evaluated based on (11), where the duty-cycle for the two active vectors ($d_i$) are calculated by solving the following equation:

$$d_i=\frac{\delta }{J_i}$$

(16)

where $\delta$ is the constant of proportionality, the subscript i denotes the adjacent vectors, in this case (i = 1; 2) and i = 0 corresponds to the duty-cycle of a zero vector, which is evaluated only one time, and $J_i$ is equivalent cost function that the sum of product of all possible two cost function combination from the three vectors cost functions. The sum of the duty-cycle associated with the two active vectors and zero vector equals to unity as:

$$d_1+d_2+d_0=1$$

(17)

By solving the previous equations, it is possible to obtain the duty-cycle for each vector as:

$$d_1=\frac{J_2{J}_0}{J_1{J}_0+J_1{J}_2+J_2{J}_0}$$

(18)

$$d_2=\frac{J_1{J}_0}{J_1{J}_0+J_1{J}_2+J_2{J}_0}$$

(19)

$$d_0=\frac{J_1{J}_2}{J_1{J}_0+J_1{J}_2+J_2{J}_0}$$

(20)

Considering these expressions, the new cost function that is evaluated at every $T_s$ for the sector selection, can be defined as:

$$g=d_1{J}_1+d_2{J}_2$$

(21)

The two vectors that minimize the CF are selected and applied to the 2L-VSI in the next sampling interval beside the zero vector. After selecting the optimal two vectors and calculating the corresponding duty-cycle, the dwell time associated with each vector can be calculated as in (22). The complete algorithm of M${}^2$PC will be executed sequentially as given in Algorithm 1.

$$\begin{array}{cc}\hfill & T_1=d_1{T}_s\hfill \\ \hfill & T_2=d_2{T}_s\hfill \\ \hfill & T_0=d_0{T}_s\hfill \end{array}$$

(22)

The conventional MPC control strategy is carried out in the following order: (1) Measuring of MPC input variables; (2) Predicting the output voltage and for the next sample period; (3) Calculating the cost function; (4) Choosing the switching signals that minimize the cost function; (5) Applying the optimized corresponding signals.

In reality, taking into consideration the computation time and applying the selected switching state after the next sample instant is a simple way to compensate for the expected delay. As a result, the control algorithm is updated in the following way: (1) Variables are measured. (2) Application of the switching state (calculated in the previous interval). (3) Estimation of the voltage values at time $t_k+1$, taking into account the switched state. (4) For all conceivable switching states, prediction of load voltages for the following sampling instant ($t_k+2$). (5) For each prediction, the cost function is evaluated. (6) The switching state that minimizes the cost function is chosen. It can be assumed that $V_{C_f}^{*}(k+1)=V_{C_f}^{*}\left(k\right)$ for the predictive voltage control example, and the cost function can be rewritten as in (15). This approximation will lead to a one-sample delay in the reference tracking of the reference currents. If compensation of the calculation time delay is considered, the reference $V_{C_f}^{*}(k+2)$ is required. Using the same idea, the future reference can be assumed to be $V_{C_f}^{*}(k+2)=V_{C_f}^{*}\left(k\right)$, resulting in the following cost function

$$\begin{array}{c}\hfill J={(V_{C_f\alpha }^{*}-V_{C_f\alpha }(k+2))}^2+{(V_{C_f\beta }^{*}-V_{C_f\beta }(k+2))}^2.\end{array}$$

(23)

Algorithm 1: Pseudocode of the conventional M${}^2$PC with the studied 2L-VSI.

3. Proposed M${}^2$PC Based-ANN for the 2L-VSI

3.1. Motivation of Using ANN in the Proposed System

The artificial neural network (ANN) can be used to model complicated systems in order to save time and effort while analyzing difficult mathematical relationships. It can map nonlinear relation between the input and output using a historical data [29]. The neuron is the fundamental unit of the neural network, as shown in Figure 3, and ANN attempts to imitate the human brain. During the training phase, the neural network attempts to map the relationship between the input and output data by changing the weighting and biasing factors, which are usually initialized randomly, by optimizing the employed cost function. By taking advantage of the flexibility of M${}^2$PC at training phase, this work proposes a radial bias function (RBF) ANN-based controller for a three-phase 2L-VSI with an output $LC$ filter targeting UPS applications. The goal is getting lower THD and good performance in the output voltage at different types of loads. The proposed controller undergoes two main steps: (i) we use it to evaluate the single objective cost function, as an expert or a teacher for generating the data required for training, validation, and testing phases in an off-line mode, where the proposed neural network applies standard supervised learning under the full-state observation of the system; (ii) when the off-line training is performed, the trained RBF-ANN can successfully calculate the duty-cycle of the two active vectors, which will be applied in the next sampling interval, and then control the output voltage of the inverter as illustrated in Figure 4. We study a performance comparison between the proposed ANN-M${}^2$PC based approach and the conventional MPC, under various operating conditions.

3.2. Description of the Implemented ANN Network

In this work, the implemented ANN network is the radial bias function, which has a structure of three layers as one input layer, one hidden layer, and one output layer. The RBF-ANN has simple design, immunity to input noise, and good generalization. These properties allow it to be used for designing flexible control systems.The RBF is utilized as an approximation function to define the active vectors’ duty-cycle, and therefore the zero vector duty-cycle can be calculated as well. As shown in Figure 4, the input layer gets the cost function for both active vectors that make up the optimum sector, as well as the cost function for the zero vector. These input features are generated from the conventional algorithm of the modulated MPC at every sampling time. The input layer distributes the network inputs into the next hidden layer. The neurons in the hidden layer, that is, $x_i$ are characterized by the Gaussian nonlinear function as in (24).

$$b_j\left(x_i\right)=exp\left(-\frac{\Vert x_i-{\mu }_j\Vert }{2{\sigma }_j^2}\right)forj=1,2,\dots ,q$$

(24)

where q is the number of neurons in the hidden layer, and ${\mu }_j$, ${\sigma }_i$, and $b_j$ are the centroid, spread width, and the response of the jth hidden neuron, respectively. The output layer combines Gaussian functions, which are weighted by weighting factors for each activation function for neurons, as in (25).

$$Y=\sum _{j=1}^q{W}_j{b}_j\left(x_i\right)$$

(25)

where Y is the output of the RBF-ANN, $b_j\left(x_i\right)$ is the output of the jth neuron in the hidden layer, and $W_j$ is the weight between the jth neuron in the hidden layer and the node in the output layer.

The dataset for RBF-ANN is selected randomly from different simulation scenarios. Different uncertainties in the following parameters are considered to collect the dataset: filter inductance and capacitance, load impedance, reference output voltage, and sampling interval as listed in

Table 1. Description of training scenarios used for training the proposed RBF-ANN.

Scenario	${\mathit{L}}_{\mathit{f}}$ [mH]	${\mathit{C}}_{\mathit{f}}$ [$\mathsf{\mu }$F]	R [$\Omega$]	L [mH]	${\mathit{T}}_{\mathit{s}}$ [$\mathsf{\mu }$s]
1	1	40	10	1	40
2	1	55	11	2	40
3	0.85	50	15	3	60
4	0.50	55	10	4	35
5	0.75	35	12	5	40
6	0.90	45	25	6	50
7	1	40	10	10	50

. The total number of raw datasets equals 5000 raws. To ensure that neuron bases may respond to overlapping regions in a search area, the design for each neuron’s propagation (${\sigma }_i$) should have a high value, keeping in mind that the threshold for neurons to react in an equal manner is not too high [30]. In the current case study, it is selected at 25, and the target goal of the RBF-ANN objective function has been achieved after 16 epochs, as shown in Figure 5.

Following the well training phase of the RBF-ANN network, the testing phase will begin to assess the trained network’s effectiveness on unknown data before merging it into the online Simulink model. The performance of the RBF-ANN network during testing with random (i.e., unseen) data is shown in Figure 6. The trained RBF-ANN will be incorporated into the FCS-MPC as shown in the flowchart in Figure 7. It is important to mention that when the sixth sector is being evaluated (i.e., $i=6$), the voltage of the second vector in this sector should be computed as the first vector (i.e., similar to lines 7–10 in Algorithm 1).

In addition, it is worth noting that when the sector of the optimal two active vectors is odd (that is, 1, 3, or 5), the switching pattern in Figure 8a should be applied to keep the switching frequency constant for one sample period. When the selected sector is even (that is, 2, 4, or 6), the switching pattern in Figure 8b should be used to establish the switching frequency. In this way, the commutation of the 2L-VSI switches will be fixed within a single sampling interval, and thus the fixed-switching operation can be achieved. Additionally, the RBF-ANN network is updated at every sampling interval with the new cost function of the active vectors and zero vector. Based on these values, new duty-cycle values will be given by the RBF-ANN output, and therefore the dwell time can be recalculated to make the converter output voltage close to its reference as possible.

4. Simulation Results and Discussion

In this section, we will show the performance of the 2L-VSI using the conventional M${}^2$PC and the proposed ANN-M${}^2$PC. To assess its performance, it is of paramount importance to check several system behaviors feeding both linear and nonlinear loads. The steady-state behavior, transient response performance, harmonic spectrum, and the total harmonic distortion of both control schemes will be investigated as well.

Table 2. Parameters of the studied 2L-VSI.

Parameter	Symbol	Value
Input voltage	$V_{dc}$	700 V
Filter inductance	$L_f$	2 mH
Filter capacitance	$C_f$	50 μF
Switching frequency	$F_{sw}$	10 kHz
Sampling time	$T_s$	100 μs
Nominal RMS output voltage (L-L)	$V_{o,ref}$	380 V

summarizes the system parameters, which were used for both control strategies, while Figure 9 shows the simulink modeling for the simulation results.

Figure 10 shows the converter output voltage ($V_{o,{\alpha }_{\beta }}$) controlled by the M${}^2$PC and the proposed ANN-M${}^2$PC feeding a linear $RL$ load. Both controllers are following a voltage reference as the only objective of their cost function. In addition, the load current is shown in Figure 11. It gives an insight on the performance of the 2L-VSI using the ANN-M${}^2$PC compared to M${}^2$PC during the instant of load change at 0.1 s. That gives a fair comparison as the available opportunity is equal for both controllers by tracking only the voltage at every sampling time based on the CF definition as in (23), but with different approaches for dwell-time calculations. It has been detected that the error from the reference of the conventional tracking algorithm is high. The error at some points in time can be spiky, resulting in a risk of stability issues based on the intended application. This outcome is verified when the error difference is calculated for both the conventional and proposed algorithms, as shown in Figure 12.

Using the proposed algorithm leads to a better power quality compared to the conventional M${}^2$PC. In addition, two important aspects have been retained by using the proposed algorithm, firstly, the algorithm can still accept adding more constraints into its operating cost function. Another aspect is that the fast response and the fixed-switching frequency are achieved, while the power quality is significantly enhanced. This is illustrated in Figure 13 by calculating the THD for both conventional and proposed algorithms at different sampling times. It is clear that the proposed approach has a superior performance compared to the conventional M${}^2$PC, as it has lower THD by around 50% in all the tested sampling intervals.

Moreover, Figure 14 and Figure 15 show how the output voltage and current at the same ground of truth have incredibly enhanced and give better shape, respectively. That is a direct result of achieving an optimal switching state, as shown in Figure 16. As it may be noticed that in the proposed ANN-M${}^2$PC, the algorithm always keeps the sector selection in a certain time span repetitive and that behavior will introduce better quality compared to the conventional M${}^2$PC.

Figure 17 highlights the switching frequency characteristics, which shows a fixed-switching operation at 10 kHz. It can be noticed that the low order harmonics in the conventional M${}^2$PC cause a high THD while, in the same zone, the proposed ANN-M${}^2$PC has lower harmonics distortion and thus avoid resonance issues.

Over and above, the controller is operating at a sampling time of 100 $\mathsf{\mu }$s, and the execution time with the conventional M${}^2$PC is calculated as 6.24 μs, while with the proposed ANN-M${}^2$PC is 9.87 μs. It is clear that the computation burden is slightly higher at the same ground truth of the converter parameters and the operating point. The algorithm is running on a personal computer employing an Intel© Core™ i5-8265U processor operating at 1.60 GHz and 16 GB of RAM.

On the other hand, the robustness of the controller at different mismatch conditions is an essential aspect. Therefore, a couple of tests, as an example, have been conducted to assess the performance of the ANN-M${}^2$PC under the mismatch of filter inductance or capacitance. It can be clearly seen from Figure 18 and Figure 19 that there is a significant enhancement in the controller robustness compared to the conventional M${}^2$PC.

Finally, Figure 20 validates that the proposed ANN-M${}^2$PC determines the duty-cycle optimally rather than with unnecessary magnitudes as obtained from the conventional M${}^2$PC. In addition, the optimal duty-cycle can be obtained by correctly training the ANN network based on accurate and diverse dataset. The benefit of the proposed system is that the FCS-MPC algorithm for UPS applications does not need to add additional terms into the cost function to improve the power quality and therefore increase the frequency of switching to higher levels. This means that the proposed idea in this paper can be applied as an explicit FCS-MPC without the need to add the voltage tracking and its derivative into the cost function to enhance the power quality.

5. Conclusions

In this paper, an ANN-M${}^2$PC control technique based on the radial bias function is proposed for a voltage source inverter with an output $LC$ filter for UPS applications. This allows the control system to improve the voltage tracking performance for both linear and nonlinear loads, resulting in low THD. The proposed technique retains all model predictive control and space vector properties, and allows a fixed-switching frequency operation. It is worth noting that the cost function has only one goal and no tunable variables. As a result, the proposed controller can provide optimum performance over a wide range of operations. In this way, the system’s steady-state and transient performance is comparable to that of a traditional M${}^2$PC, but with significantly better power quality and a little increase in computational burden.