A Neural-Network-Based Model Predictive Control of Three-Phase Inverter With an Output Filter
Model predictive control (MPC) has become one of the well-established modern control methods for three-phase inverters with an output filter, where a high-quality voltage with low total harmonic distortion (THD) is needed. Though it is an intuitive controller easy to understand and implement, it has the significant disadvantage of requiring a large number of online calculations for solving the optimization problem. On the other hand, the application of model-free approaches such as artificial neural network-based (ANN-based) approaches is currently growing rapidly in the area of power electronics and drives. This paper presents a new control scheme for a two-level converter based on combining MPC with feed-forward ANN, with the aim of getting lower THD and improving the steady and dynamic performance of the system for different types of loads. First, MPC is used, as an expert, in the training phase to generate data required for training the proposed neural network. Then, once the neural network is fine-tuned, it can be successfully used online for voltage tracking purpose, without the need of using MPC. The proposed ANN-based control strategy is validated through simulation, using MATLAB/Simulink tools, taking into account different loads conditions. Moreover, the performance of the ANN-based controller is evaluated, on several samples of linear and non-linear loads under various operating conditions, and compared to that of MPC, demonstrating the excellent steady-state and dynamic performance of the proposed ANN-based control strategy.
The three-phase inverter is an extensively popular device, which is commonly used for transferring energy from a DC voltage source to an AC load. The control of three-phase inverters has received much attention in the last decades both in the scientific literature and in the industry-oriented research [1, 2]. In particular, for applications such as uninterruptible power supplies (UPSs), energy-storage systems, variable frequency drives, and distributed generation, the inverters are commonly used with an output filter to provide a high-quality sinusoidal output voltage with low total harmonic distortion (THD) for various types of loads, especially for unbalanced or nonlinear loads [3, 4, 5, 6, 7]. However, the performance of the inverter is mainly dependent on the applied control technique. These controllers must cope with the load variations, the non-linearity of the system, and ensuring stability under any operating condition with a fast transient response .
In the literature, various types of classical and modern control schemes have been studied and proposed in order to improve the performance of the converters, such as non-linear methods (e.g., hysteresis voltage control (HVC)) , linear methods (e.g., proportional-integral (PI) controller with pulse-width modulation (PWM) and space vector modulation (SVM)) [10, 11, 12, 13], multi-loop feedback control [14, 15], deadbeat control [16, 17, 18, 19, 20], repetitive-based controllers [21, 22], linear quadratic controller (LQR) , and sliding-mode control [24, 25]. Most of these control schemes, in a way or another, are characterized by a number of limitations.
Just to name a few, the major drawback of non-linear methods (e.g., HVC), which require high switching frequency for effective operation, is having a variable switching frequency which creates resonance problems which reduce the converter’s efficiency [26, 27]. On the other hand, although the linear methods, which require carrier-based modulators, have the advantage of constant switching frequency, their dynamic response is weak comparing with HVC, because of the slow response of the modulator. However, both linear and nonlinear methods are extensively used for generating the switching signals of the inverter because of the simplicity of the controller implementation. Another example is deadbeat control which provides fast transient response, however, it has high sensitivity to model uncertainties, measurement noise, and parameter perturbations, in particular for high sampling rates. Other modern control approaches based on control theory  and synthesis  have been proposed, with the aim of handling the possible uncertainties in the system.
Model predictive control (MPC) has become one of the well-established modern control methods in power electronics, particularly for three-phase inverters with filter according to [26, 1, 30, 31, 32]. The key characteristic of MPC is to explicitly use the model of the system to predict the future behavior of the variables to be controlled, considering a certain time horizon. Afterwards, MPC selects the optimal control action (i.e., optimal switching signals) based on the minimization of a pre-defined cost function, which represents the desired behavior of the system [33, 34, 35]. The main features of MPC can be summarized as: (i) an intuitive controller easy to understand and implement, with a fast dynamic response; (ii) the simple inclusion of system constraints and nonlinearities, and multivariable cases; (iii) the flexibility to include other system requirements. On the other hand, a major drawback of MPC is that it requires the optimization problem to be solved online, which involves a huge amount of real-time calculations. However, different solutions have been introduced in order to address this problem, as proposed in [36, 37, 31].
On the other hand, the application of data-driven methodologies (or model-free approaches, particularly artificial neural networks ANNs-based approaches) is currently growing rapidly in the area of power electronics and drives . Broadly speaking, the use of neural networks for the control of dynamical systems was proposed in the early nineties [39, 40, 41]. Multi-layer perceptrons were employed in various roles, including system identification and implementation of the control law. In particular, ANN-based controllers and estimators have been widely used in identification and control of power converters and motor drives . As an example, they can be used to estimate the rotor speed, rotor-flux, and torque of induction motors [43, 44, 45], in addition to the identification and estimation of the stator current of induction motor drives . Moreover, several ANN-based methods have been used in control of power converters, as presented in [47, 48, 49, 50]. Indeed, the ANN-based controllers have some advantages compared to other control methods such as: (i) their design does not require the mathematical model of the system to be controlled, considering the whole system as a black-box; (ii) they can generally improve the performance of the system when they are properly tuned; (iii) they are usually easier to be tuned as compared to conventional controllers; (iv) they can be designed based on the data acquired from a real system or a plant in the absence of necessary expert knowledge. But, they require a large number of training data. However, as the present work suggests, this is not a major drawback because large amounts of data can be obtained using reliable simulation tools.
By taking advantage of the flexibility of MPC at training time, this paper proposes a feed-forward ANN-based controller for a three-phase inverter with output filter for UPS applications, with the aim of getting lower THD and good performance for different types of loads. The proposed controller undergoes two main steps: (i) the use of MPC as an expert or a teacher for generating the data required for training off-line the proposed neural network using standard supervised learning, under full-state observation of the system; (ii) once the off-line training is performed, the trained ANN can successfully control the output voltage of the inverter, without the need of using MPC at test time, as illustrated in Fig. 1. A performance comparison between the proposed ANN-based approach and the conventional MPC, under various operating conditions, is studied. The main contributions of the work described in this paper can be summarized as follows:
To the best of our knowledge, this is the first attempt to directly control a three-phase inverter with an output filter using a feed-forward ANN based on MPC, instead of the more common model-based approaches as well as ANN classical control-based (such as Fuzzy Logic Controller FLC-, PID-, or PWM-based) approaches, or a combination of both [48, 51, 52, 53, 54, 55].
The proposed ANN-based approach generates directly the switching signals of the inverter, without the need for the mathematical model of the inverter and without a pre-defined cost function to be minimized at each sampling time .
An open repository of the dataset and codes111Web: https://github.com/IhabMohamed/ANN-MPC is provided to the community for further research activities.
The rest of the paper is organized as follows. Section II deals with the mathematical model of the three-phase voltage-source inverter with filter, whereas in Section III the proposed predictive controller strategy is explained. The ANN-based control scheme proposed in this paper is described in Section IV. In Section V, simulation implementation and results are discussed for both proposed control schemes, then the conclusion is provided in Section VI.
Ii System Description and Modeling
In this section, the mathematical interpretation of the converter system considered in this paper is presented. Moreover, the model of filter is described in details, and is then used by the predictive controller to predict the output voltage for all given input voltage vectors.
Ii-a System description via Clarke transformation
The power circuit of the three-phase voltage-source inverter considered in this paper is depicted in Fig. 2. In the present case, the load is assumed to be unknown, while the models of the converter and filter are presented . Moreover, the two switches of each leg of the converter operate in a complementary mode, in order to avoid the occurrence of short-circuit conditions. Thus, the switching states of the converter can be represented by the three binary switching signals, , , and , as follows:
These switching states can be expressed in vectorial form (i.e., in reference frame) by following transformation:
where . It is noteworthy that the switching devices are assumed to be ideal switches, therefore the process of switching-ON/-OFF is not taken into consideration .
The possible output-voltage space vectors generated by the inverter can be obtained by
where , , and represent the phase-to-neutral, , voltages of the inverter. On the other hand, the voltage vector can be defined, in terms of the switching state vector S and the dc-link voltage , by
Fig. 3 illustrates the eight switching states and, consequently, the eight voltage vectors generated by the inverter using (1) and (3), considering all the possible combinations of the switching signals , , and . It is noteworthy that only seven different voltage vectors are considered as possible outputs, since .
Similarly, as in (1), the filter current , the output voltage , and the output current can be expressed in the vectorial form as
Ii-B filter modeling
As far as the model of filter is concerned, it can be described by two equations, the former describes the inductance dynamics, whereas the latter describing the capacitor dynamics . These two equations can be written as a continuous-time state-space system as
where and are the filter inductance and the filter capacitance, respectively. The output voltage and the filter current can be measured, whilst the voltage vector can be calculated using (3). The output current is considered as a disturbance due to its dependence on an unknown load, whereas the value of is assumed to be fixed and known. The output voltage is considered as the output of the system, which can be written as a state equation as .
Then, using (7), the discrete-time state-space model of the filter can be obtained for a sampling time as
_x(k+1) = ⏟ e^AT_s_A_q ⏟ [if(k)vc(k)]_x(k) + ⏟∫^T_s_0 e^AτB dτ_B_q v_i(k) + ⏟∫^T_s_0 e^AτB_d dτ_B_dq i_o(k).
This model is used by the predictive controller (i.e., MPC) to predict the output voltage for all given input voltage vectors . Then, for predicting the output voltage using (II-B), the output current is needed and can be estimated using (8), assuming that for sufficiently small sampling times as proposed in [1, 35].
Iii Model Predictive Control for Neural Network
In this section, the model predictive control (MPC) proposed in [30, 34], and which provides the state-of-art of output-voltage control of three-phase inverter for UPS applications, has been used: (i) to generate the data required for the off-line training of the proposed neural network, and (ii) to compare its performance with the proposed ANN-based controller under linear and non-linear load conditions.
Iii-a Proposed Predictive Controller Strategy
In the proposed control strategy, it has been assumed that the inverter generates only a finite number of possible switching states and their corresponding output-voltage vectors, making it possible to solve the optimization problem of the predictive controller online . MPC exploits the discrete-time model of the inverter to predict the future behavior of the variables to be controlled, for each switching state. Thereafter, the optimum switching state is selected, based on the minimization of a pre-defined cost function, and directly fed to the power switches of the converter in each sampling interval , without the need for a modulation stage. The cost function to be minimized has been chosen in order to achieve the lowest error between the predicted output voltage and the reference voltage. In this work, a cost function which defines the desired behavior of the system is expressed in orthogonal coordinates by
where and are the real and imaginary parts of the output-voltage reference vector , while and are the real and imaginary parts of the predicted output-voltage vector .
The block diagram of MPC, considering only one-step prediction horizon, for a three-phase inverter with output filter is shown in Fig. 4. The control cycle of the predictive controller at sampling instant is described as a pseudo code in Algorithm 1 with more detail. Line of the code declares the control function, where the switching signals , , and are the outputs, while the inputs are the measured variables of the filter current , the output voltage , and the reference voltage at sampling time , all expressed in coordinates. The two variables, and , are recalled from the previous sampling instant (lines to ), which are firstly initialized for (lines to ). These two variables are used to estimate the output current given by (8) (line ), in order to obtain the possible predictions of using (II-B).
The optimization problem to be solved is performed between lines and . The code sequentially selects one of the seven possible voltage vectors generated by the inverter based on (3) (line ) and applies it, in order to obtain the output voltage prediction at instant , as in line . The cost function given by (9) is used to evaluate the error between the reference and the predicted output voltage at instant for each voltage vector (line ). The code selects the optimal value of the cost function , and the optimum voltage vector is then chosen (lines to ). Note that, is initialized with a very high value (line ). Finally, the switching states, , , and , corresponding to the optimum voltage vector are generated and applied at the next sampling instant (line ), as illustrated in Fig. 3.
In fact, all the control approaches proposed in the literature, in a way or another, are model-based approaches, which require in general either diverse computational or approximative procedures for applying their solution. In this context, MPC, the widely used approach for three-phase inverters, relies on solving an optimization problem online, leading to a large number of online computations. In other words, the control signal of MPC is determined by minimizing a cost function online at each time instant. Moreover, recently the artificial neural networks have been used in conjunction with MPC, in order to provide a powerful and fast optimization as proposed in [56, 57, 58, 59].
The alternative approach to be considered in the present work is to apply neural network-based function approximators, which can be trained off-line to represent the optimal control law. Such an approach is expected to avoid the drawbacks associated with MPC-based control approaches, does not require the mathematical model of the system to be controlled, does not evaluate a cost function online at each sampling time, and, therefore, does not rely on an optimization problem to be solved online. For this reason, this paper focuses on the control of a three-phase inverter with output LC filter using a feed-forward ANN-based MPC, which has not been reported in the literature, where MPC is only used as a teacher for training the neural network.
Iv Implementation of ANN-Based Controller
In this section, some important concepts related to ANN including the structure of the proposed ANN-based controller as well as details on the training data will be covered.
Iv-a Proposed Neural Network Architecture
Machine learning, and in particular artificial neural networks, is one key technology in modern control systems. An artificial neural network is an extremely flexible computational model that can be optimized to learn input-to-output mappings based on historical data. This model can be expressed as
where is an activation function (usually it is a non-linear function such as logistic sigmoid or hyperbolic tangent, to ensure the universal approximation property ), is the input vector of the ANN with elements, are the weights for each input , and is a bias or correction factor. Indeed, the objective of the ANN training phase is to optimize and . Although the most recent developments have focused on larger and larger scale problems (deep learning), improved techniques have also been proposed to improve the reliability of networks of smaller size. The result is a sound and flexible technology.
An artificial neural network (ANN) is composed of a number of simple computing elements organized in layers and linked by weighted connections. Feed-forward networks do not contain loops, so they have a static behavior and can be used to implement memoryless input-to-output mappings. In a feed-forward network it is possible to distinguish one input layer, one output layer, and hidden layers that connect the input to the output.
In this work, a feed-forward neural network (fully connected multi-layer perceptron) of the “shallow” type, i.e., one hidden layer, was used to implement the control model. A grid search tuning procedure allowed the selection of a configuration with 15 units in the hidden layer, while the number of input and output units is constrained by the number of input and output variables, respectively. Training was done via the Scaled Conjugate Gradient (SCG) method , which exploits the good convergence properties of conjugate gradient optimization  and has the computational advantage of not requiring a line search, nor any user-selected parameters.
Iv-B ANN Training Procedure
The ANN takes as inputs the measured variables of the filter current , the output voltage , the output current , and the reference voltage all expressed in coordinates. For simplicity’s sake, the real and imaginary parts of these variables are separately fed to the neural network, bringing the total number of input features to eight, i.e., . On the other hand, the optimum voltage vector to be applied at each sampling instant constitute the output of the ANN. In fact, the size of the output layer is an array with a length of , which represents the indexes of the seven possible voltage vectors that inverter generates. The output is one-hot encoded, meaning that at each sampling instant only the index of the optimum voltage vector will be active (i.e., having a value of one), while others will be equal to zero.
The training data, which have been collected by MPC, comprises samples, which are divided into samples for specific resistive loads (i.e., for only and ), whereas only samples represent the case where the inverter is directly fed a non-linear load (i.e., diode-bridge rectifier) with different values of and . For each sample, the simulation is run, using MPC222Web: https://github.com/IhabMohamed/MPC-3-Phase-Inverters, under various operating conditions such as simulation time (i.e, number of output voltage cycles), sampling time , filter capacitor , filter inductance , DC-link voltage , and reference voltage . Then, the input features of the neural network and their targets are stored for training.
As a consequence, the total dataset consists of and instances for the case of having and training samples, respectively. These dataset has been divided into two parts: randomly selected for training purposes, and for testing and validation. The overall accuracy of ANN in the case of -samples is , while it has a increase for the -samples case, considering hidden layers and the training function “transcg”. It is observed that the validation and training error, as well as the error on the test set, are very similar when training stops, according to the “early stopping” criterion used. This is an indication that the neural network may attain a good degree of generalization. For instance, for the -samples case, the best validation performance is taken from epoch with the lowest validation error of . The training results are summarized in Table I. It is noteworthy that training was also done using Bayesian regularization backpropagation method, achieving an accuracy of . However, its performance at the test phase (on-line) was not satisfactory.
|Tr. Samples||No. of Instances||Accuracy||Validation Error (epoch)|
For further detailed information about the training samples used for training the ANN-based controller, please refer to: https://github.com/IhabMohamed/ANN-MPC.
Iv-C ANN-Based Controller
As previously mentioned, the ANN-based controller is trained off-line from samples collected via MPC, as shown in Fig. 1. After fine-tuning the ANN, the trained ANN can be used instead of MPC to control the system presented in Fig. 2.
Fig. 5 depicts the proposed block diagram of the ANN-based controller for a three-phase inverter with output filter, in order to generate a high-quality sinusoidal output voltage with low THD, considering different types of loads.
The control strategy of the proposed ANN-based controller at sampling time can be described as follows:
then, these measured values in addition to the reference voltage are used by the trained ANN in order to explicitly generate the optimum voltage vector to be applied at instant ;
finally, the switching states, , , and , corresponding to the optimum voltage vector are applied and directly given to the power switches of the converter each sampling interval .
V Simulation Implementation and Results
This section provides a comprehensive study and evaluation of the two proposed control strategies, taking into account different loads under various operating conditions.
V-a Simulation Setup
The Simulink model and the simulations of the system shown in Fig. 2 have been implemented using MATLAB (R2018a)/Simulink software components, which runs on Ubuntu 16.04 bit, in order to verify the proposed ANN-based control strategy and compare its performance with the conventional predictive controller (i.e., MPC). A lower-end PC has been used for acquiring the training samples, off-line training, and online voltage tracking purpose using the proposed ANN approach. In particular, it is equipped with an Intel® Core i5-4210U CPU, GB of RAM, and an Nvidia Geforce® GPU, and runs Ubuntu 16.04 bit.
V-B Simulation Results
The simulation of the three-phase inverter system shown in Fig. 2 was carried out, considering linear (i.e., resistive) and non-linear loads, in order to evaluate the behavior of the proposed ANN-based control strategy and compare its performance with that of MPC proposed in Section III. In particular, the steady and dynamic performance of both control strategies are studied and evaluated, taking into account different loads conditions. The parameters of the system are listed in Table II.
The behavior of the ANN-based controller in steady-state operation for a resistive load of shown in Fig. 6, while the behavior of the predictive controller for the same resistive load is shown in Fig. 7. It is noteworthy that the amplitude and the fundamental frequency of reference voltage are set to and , respectively. It can be seen in the figures that the output voltages for the proposed control strategies are sinusoidal with low distortion, particularly for the ANN-based approach which has a THD of only compared to for MPC. Moreover, it is observed, due to the resistive load, that the output current is proportional to the output voltage, whilst the filter current measured at the output of the converter shows high-frequency harmonics, especially in the case of MPC, which are attenuated by the filter.
The transient response of both the control strategies for no-load (i.e., open-circuit) is shown in Fig. 8 and Fig. 9. Here, the filter capacitor and filter inductance are set to and , respectively, whilst the sampling time is kept constant at a value of . It can be seen that the ANN-based controller permits a fast and safe transient response, demonstrating the excellent dynamic performance of the proposed ANN-based control strategy. For MPC, the time elapsed in order to reach steady-state operation and to faithfully track its reference waveform is about ( cycle), which is affected by the change in the load, as illustrated in Table III. On the other side, for the ANN-based controller, it is observed that it takes less than for any load, in order to reach steady-state. Furthermore, the output voltage quality of ANN-based approach is improved significantly, with a THD of compared to for MPC.
As previously mentioned, the proposed ANN is trained off-line using a dataset which represents only different values of resistive load under different operating conditions. However, in order to verify the feasibility and effectiveness of the proposed ANN-based controller under realistic conditions, the behavior of the system is tested online considering non-linear loads, such as a diode-bridge rectifier as shown in Fig. 10 and an inductive load. Fig. 11 and Fig. 12 show the behavior of the proposed control strategies for a diode-bridge rectifier, with values and , while the behavior for an inductive load of is shown in Fig. 13 and Fig. 14, considering the same operating conditions presented in Table II and different amplitudes of the reference output voltage. As can be seen in the figures, the output voltage generated by the ANN-based controller outperforms that obtained using MPC for non-linear loads, despite the highly distorted output currents due to feeding a non-linear load. For instance, for MPC, the total distortion in the output voltage for the inductive load was , while it was for the ANN-based controller. The result of MPC can be improved by using either a smaller sampling time or a higher value of the filter capacitance .
In order to achieve a fair comparison and prove the superiority of the proposed ANN-based approach compared to MPC in both transient and steady-state response, Table III shows a comprehensive comparison of both the control strategies for linear and non-linear loads, under various operating conditions such as sampling time , filter capacitor , filter inductance , DC-link voltage , and reference voltage . Fifty unseen samples, at training time, have been considered for testing the proposed approaches, thirty samples for different values of a resistive load, whereas the rest for a diode-bridge rectifier as a non-linear load. Moreover, the THD of the output voltage obtained by the proposed control strategies, for some samples given in Table III, is visualized in Fig. 15. As anticipated, the performance of the ANN-based approach, either based on sixty or seventy training samples, outperforms that of MPC, which can be noticed in lower THD and less settling time to reach steady-state (i.e., , as shown in the first ten samples (i.e., )). It can be noticed that the performance of the ANN-based controller using only sixty training samples is similar to that based on seventy samples (see column and in Table III).
However, for samples , the output voltages obtained using MPC are better than that obtained using the ANN-based controller. Moreover, it can be seen in sample that the ANN-based approach failed to control the output voltage and track its reference waveform. As a consequence, the UPS does not work properly due to a higher distortion in the voltage. These results could be improved using either (i) a higher sampling frequency, or (ii) a higher value of the filter capacitance , as illustrated in sample which represents an improvement of the result of sample . An alternative solution to be considered to improve the performance of the controller is to increase the number of training instances, taking into account various values of and . In addition, it is observed that having a one-delay step in the input features of the neural network improves its performance to outperform that of MPC. For example, of samples is decreased to be , respectively.
In fact, it is not surprising that the performance of the proposed ANN-based controller outperforms that of MPC in both transient and steady-state response, even with unseen samples (i.e., loads) at training time as tabulated in Table III. This happened for two reasons. First, the training data are sufficient to learn the mathematical model of the system to be controlled and its dynamics, as well as representing the optimal control law. Second, generating a sinusoidal output voltage can be considered as a repetitive task, where neural network can easily detect and learn repetitive sequences of actions.
|Case # : Resistive Load as Linear Load with||Results|
|Case # : Diode-Bridge Rectifier as Non-Linear Load with and||Results|
At the moment, one can say that the main limitation of the proposed method is that only the simulation results are not sufficient to prove its novelty in practical applications. However, indeed we believe that our proposed approach will also represent a novel contribution to the practical applications for the following reasons: (i) based on the previously proposed literature, both ANN-based and MPC-based approaches have shown good results in both simulated and experimental scenarios; (ii) moreover, the trained network is only required to be fine-tuned, in order to improve its performance in practical applications.
Vi Conclusions and Future Work
A novel control strategy using a feed-forward ANN to generate a high-quality sinusoidal output voltage of a three-phase inverter with an output filter has been successfully developed and tested, for different types of loads under various operating conditions. The output voltage of the inverter is directly controlled, without the need for the mathematical model of the inverter, considering the whole system as a black-box. In this work, MPC has been used for two main purposes: (i) generating the data required for the off-line training of the proposed ANN, and (ii) comparing its performance with the proposed ANN-based controller for linear and non-linear load conditions. Simulation results, based on fifty test samples different than those that were used at training time, show that the proposed ANN-based controller performs better than MPC, in terms of a lower THD and a fast and safe transient response, demonstrating the excellent steady and dynamic performance of the proposed ANN-based control strategy. As in any model-based control strategy, variations in the system parameters inevitably influence the performance of the ANN-based control scheme proposed in this paper. The possible directions for future work would be (i) the implementation of the ANN-based controller in practical applications; then (ii) the employment in other power electronics applications, possibly employing different neural networks.
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