An improved low-computation model predictive direct power control strategy for three-level four-leg active power filter

Model predictive direct power control (MPDPC) strategy offers numerous benefits over the traditional control strategies in active power filters (APF). However, the real-time execution duration (RTED) reduction of model predictive control in multilevel inverters is a big challenge. In this paper, an improved low-computation model predictive direct power control strategy for three-level four-leg APF (TL-FL-APF) is presented. First, based on generalized instantaneous reactive power theory, a new predictive power model for TL-FL-APF is established. Then, in order to reduce the RTED, classical multiple power predictions are converted into single voltage prediction through the principle of deadbeat control, and the optimized vector range is reduced from 81 to 4–18 by the method of spatial stratification using the γ-component of reference voltage vector. Finally, the simulation and experimental results demonstrate that the proposed low-computation MPDPC has excellent performance.


Introduction
In recent years, due to the extensive use of nonlinear and unbalanced loads, the power quality of three-phase four-wire power system has become increasingly serious. Four-wire APF is an ideal device to solve the problem of current distortion and excessive neutral-wire current [1][2][3], and the common topology of four-wire APF can be divided into three-leg and four-leg. Four-leg APF has strong neutral-wire current compensation ability, simple dc-link voltage control, and low capacitance requirement [4,5], so it has a broader application prospect. Three-level APF (TL-APF) provides superior performance, e.g., low dv/dt of output voltage, low ripples, low harmonic content current, etc., as compared to two-level APF [6]. Therefore, TL-FL-APF is the best option to deal with distorted/unbalanced grid currents.
TL-FL-APF was first presented in [7], and plentiful research focuses on developing control strategies of FL topology. The classic control strategy of FL-APF is vector B Zhan Liu liuzhan_cumt@163.com 1 Department of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou, China control based on 3D-SVPWM combined with linear controller, such as repetitive control and proportional resonance control [8][9][10]. Moreover, in [11,12], the sliding-mode control and passivity-based control are proposed.
The difficulty of tuning parameters and the complexity of pulse modulation are its obvious shortcomings for [8][9][10][11][12]. The finite control set model predictive control (FCS-MPC) is a control strategy based on discrete mathematical model of system. Without pulse width modulation block, flexible control and fast response are its significant advantages [13][14][15][16]. This technique has been successfully applied for power inverters [17], matrix converters [18,19], and motor drive applications [20,21].
The predictive controllers of APF are mainly divided into two categories: model predictive current control (MPCC) and MPDPC [22]. Direct harmonic power detection is simpler than harmonic current detection, avoiding the use of phaselocked loops and dq coordinates [23,24]. There are two main problems in TL-FL-APF based on predictive control: Firstly, the number of basic voltage vectors brings heavy prediction computation [25] and secondly, TL-APF needs to ensure the balance of the neutral point while considering the power following [26].
In the classical MPDPC of TL-FL-APF, APF needs to complete 81 times of power prediction and neutral-point potential prediction and cost function optimization; meanwhile, weight factor is also time-consuming task [27]. Three virtual mutually orthogonal sliding planes were proposed in abc coordinates to lower calculation burden in [28]. Its alternative base voltage vectors are reduced from 81 to 3, but the compensation performance becomes relatively worse. The power prediction model of multilevel four-leg grid connected converter was proposed for the first time in [29]; moreover, ρ-σ-τ coordinates were proposed to identify the type of tetrahedron and select the appropriate voltage sequence. The judgment of large sectors and the calculation of vector action time have been simplified. However, zero-sequence reactive power part of power prediction model has limitations, and [29] does not consider the delay compensation of model predictive controller. Furthermore, due to the excellent parallel data processing ability of FPGA, there is a hardware solution to handle computational burden of MPC. Gulbudak and Santi [30] and Martin Sanchez et al. [31] accordingly proposed predictive controllers for direct matrix converter and power converters based on FPGA.
Taking a cue from the literature survey presented in the preceding paragraphs, in this article, an improved lowcomputation MPDPC based on a new predictive power model is proposed. Low-computation MPDPC aims to exploit the space vector distribution in αβγ coordinates to reduce the RTED of predictive controller. Based on the principle of deadbeat control, multiple power predictions are converted into single voltage prediction. Based on the method of spatial stratification, 81 voltage vectors are divided into 13 categories. Based on the γ-component of reference voltage vector, optimized vector range is reduced from 81 to 4-18.
The contribution of this article is summarized as follows: 1. Proposed a new power model for FL-APF, and considering delay compensation method is given correspondingly. 2. Proposed a new spatial stratification method to reduce the computational burden of predictive controller.
The remaining of this article is organized as follows. In Sect. 2, discrete-time power predictive model for TL-FL-APF is established. The deficiencies of the zero-sequence reactive power prediction model and the solution for delay compensation are given. In Sect. 3, the core concept of low-computation MPDPC strategy is explained, which comprises reference voltage calculation, space stratification, and neutral-point balance. Section 4 includes simulation and experimental validation of the proposed strategy. Finally, Sect. 5 concludes this article.

Power predictive model
TL-FL-APF with an output L filter considered in this article is shown in Fig. 1. It presents a connection format similar to conventional three-leg APF, with an additional leg connected to the neutral wire of four-wire grid.
The dynamic equations of TL-FL-APF output current i αβγ in the αβγ reference frame can be expressed as the function of the output converter voltage v αβγ and grid voltages e αβγ as follows: Using the forward Euler method to discretized (1), the current predictive model is given as follows: The output voltages v αβγ of APF can be expressed as follows: where S A,B,C,N are the switching functions of the TL-FL-APF and U dc is the single bus capacitor ideal voltage. Based on generalized instantaneous reactive power theory, the compensation of FL-APF is given as follows: where p,q, and q 0 are the active, reactive, and zero-sequence reactive powers, respectively, of the TL-FL-APF in the αβγ coordinates.
The derivatives of the APF powers defined in (3) are given as follows: By neglecting the change in the grid voltages over a sampling period T s and using (1), the power model is obtained as follows: Using the forward Euler method to discretized (6), the final power predictive model is given as follows:

Neutral-point voltage predictive model
The upper and lower bus capacitor voltages are U dc1 and U dc2 , respectively, and the neutral-point potential Δu is defined as Δu U dc1 − U dc2 . The differential of Δu can where C1, C2 C and i o is the neutral-point current.
Using the forward Euler method to discretized (8), the neutral-point potential predictive model is given as follows:

Calculation of reference power and delay compensation
This section describes reference power computation technique. The main goal of MPDPC is that powers p, q, and q 0 will be forced to track their references p*, q*, and q0*. As shown in Fig. 2, after Clarke transformation of grid voltage and load current, the instantaneous power calculation is given as follows: Active power and reactive power are composed of DC power and harmonic power, p Lf , q Lf can be filtered out through the low-pass filter, and then, p* and q* can be obtained. The product of (e α -e β ) and zero-sequence load current i Lγ is defined as zero-sequence reference reactive power q* 0 [32].
Considering the delay compensation problem of FCS-MPC, calculate the reference power at the (k + 2)th instant utilizing the Lagrange extrapolation, and it can be expressed as follows: However, there are certain flaws in the power prediction model. Assuming that grid voltage is ideal sine wave and phase voltage amplitude is E, after invariant power Clarke transformation, (e α -e β ) is obtained as follows: There are two zero-crossing points of (e α -e β ) in each fundamental cycle, and APF will completely lose the compensation ability of q0. If reference delay compensation of q* 0 is performed, the tracking of q 0 cannot be guaranteed and will deteriorate compensation effect of APF. Therefore, this article considers reference delay compensation of p* and q*. Meanwhile, it considers q* 0(k + 2) q* 0(k).

Control delay compensation
First, the control delay compensation for one sampling period is performed based on power prediction model [33]. Substitute the predictive power at time k + 1th instant into the power prediction model as a known quantity, and the predicted power at time (k + 2)th instant can be expressed as follows: The extrapolation of Δu is necessary to be performed as follows: where Δu(k + 1) and i x (k + 1) can be calculated by (9) and (2), respectively.

2.5. Cost function
Classical MPDPC of TL-FL-APF has two objectives: The first is to achieve a fast and accurate power tracking and the second is to realize the neutral-point potential balancing by using weighting factors. Thus, the cost function of classical MPDPC based on the power predictive model is expressed where λ is the weighting factor, which sets the relative importance of power tracking and neutral-point balancing.
It can be seen that the selection of optimal switching state for TL-FL-APF requires 81 power predictions in classical MPDPC, 81 neutral-point voltage predictions, and 81 cost function evaluations. This means that computational burden is excessive.

Determination of reference voltage vector
According to the principle of deadbeat control, replace the predicted power with the reference power at the (k + 2)th instant, and the equivalent reference voltage vector can be obtained as follows: The further solution of v α , v β is expressed as follows: According to the deviation of voltage vector, the cost function is defined as follows:

Fig. 3 Spatial distribution of voltage vector in αβγ coordinate
The cost function J 1 deals with the minimization of voltage tracking error, and all switch states of TL-FL-APF are involved in optimization.
In each sampling period, the power prediction calculation of classical MPDPC is reduced from 81 to 1. However, the number of comparison operations for voltage vectors still has 81 times.

The method of space stratification
The spatial distribution of voltage vector of TL-FL-APF is shown in Fig. 3. In αβγ coordinates, the endpoint of 81 basic voltage vectors of TL-FL-APF is distributed on 13 The maximum amplitude of vector is 2U dc . In addition to zero vector of the layer O, other vector whose two vector coordinates overlap is called as redundant vector, and TL-FL-APF contains 14 pairs of redundant vectors.
The method of spatial stratification refers to the reference voltage vector obtained by formulas (16)(17), combined with   The judgment basis of M is shown in Fig. 4, and compare γ-axis component of reference voltage vector v* with the value of ±U dc / In Fig. 5, v* is located between AO layer and M 1. Based on the principle of selecting the closest plane from v*, all switch states of AO layer form the set of switch states S. When v* is located at different positions, layer M, the constitute plane of S, and the number of elements R are summarized in Table 1.
In order to avoid phase voltage jump, the switch state (S a (k), S b (k), S c (k), S n (k)) in the upper cycle and the switch state (S a (k + 1), S b (k + 1), S c (k + 1), S n (k + 1)) in the next cycle should satisfy the constraints: v* has the same randomness as reference power, and there may not exist a suitable switch state in S. Therefore, the transition layer is set to realize the transition of switch state. The followability of power is appropriately sacrificed to meet phase voltage jump.
When v* γ (k) was located in FE layer in previous period, the low-computation MPDPC determines the optimal switch state from S {pppn, oppn, ppon, popn} and acts on the next sampling period. If v* γ (k + 1) is located in E -F − layer, lowcomputation MPDPC determines the optimal switch state from S {nonp, onnp, nnop, nnnp}, and A phase is prone to voltage jumps. If OA − layer is set as transition layer, both {oppp, oono, onoo} and {oooo, opno, onpo} meet A phase voltage jump limitation, further selecting the vectors satisfying the voltage jump of other phase constraints. On the contrary, if v* γ jumps from E -F − layer to FE layer, AO layer is set as transition layer, and it can be expressed as follows:  Fig. 6 Analysis of neutral-point potential shift

Neutral-point potential balance
The corresponding relationship of neutral-point current and redundant vectors of TL-FL-APF is listed in Table 2. In Fig. 6, solid arrow is the reference direction of neutral-point current i o . If U dc1 > U dc2 , this requires that the neutral-point current flows in the direction of the dashed arrow. If U dc1 < U dc2 , this requires that neutral-point current flows in the direction of solid arrow. According to this principle, the selection criteria of 14 pairs of redundant vectors can be determined as given in Table 3.
As depicted in the block diagram of the control scheme (Fig. 7), the control system consists of four parts: coordinate transformation, reference harmonic power calculation, APF compensating power calculation, and low-computation MPDPC.  The overall process of low-computation MPDPC is shown in Fig. 8. First, relevant power, voltage, and current information of system is sampled. Reference and control delay compensation are performed, and based on the principle of deadbeat control, v* is obtained. According to the method of spatial stratification, and combined with transition layer, M and S are determined, and then, based on cost function of voltage followability, optimization is determined. The redundant vector selection and phase voltage jump constraints of are considered. Finally, the optimal switching state S opt is determined.

Qualitative analysis of control performance
Classical MPDPC selects the switching states with the best power followability among the switching states of TL-FL-APF. Actually, it essentially finds the base voltage vector that is closest to v*. In the low-computation MPDPC, the optimal state can only be guaranteed to be optimal in S, but it cannot be guaranteed to be optimal in all switching states.
Assuming that v* is located at AO layer, (a, b) are the coordinates of the endpoints of v*. Compress the basic voltage vectors of B, A, and O layers to the αγ plane, and the basic voltage vectors with different β components and the same αγ component are abbreviated as the same point. The corresponding switching states of the basic voltage vector at each point are marked in Fig. 9. When v* is located in the position as shown in Fig. 9, the classical MPDPC can select optimal voltage vector which has the shortest distance from v* to voltage vectors between points P and K. Low-computation MPDPC adopts the idea of space stratification to determine all the basic voltage vectors of AO layer to participate in optimization and find optimal basic voltage vector among them. However, the basic voltage vector of point P is not included. If the distance of PG is shorter than the distance of KG, voltage vector selected by the classical MPDPC and low-computation MPDPC will be different.
Assuming that the β component of v* has the same distance from points P and K, the cost function in the αγ plane can be redefined as: The geometric meaning of J is to calculate the square of the distance between the endpoint of the basic voltage vector and the endpoint of the v* in the αγ plane. Then, the corresponding cost functions J K and J P of points P and K are as follows: As shown in Fig. 10, when the distance of PG is shorter than the distance of KG, it can be equivalently expressed by the cost function as follows: When v* falls on AO layer, its constraints are as follows: Combining (23) and (24) to perform linear programming of spatial area on αγ plane, the shaded area is that satisfies the constraints in Fig. 10. When the endpoint of v* falls in the shaded part, the optimal basic voltage vector determined by classical MPDPC is the basic voltage vector at point P, while the optimal vector determined by low-computation MPDPC is the basic voltage vector at point K. At this time, the compensation effect of classical MPDPC will be better than that of low-computation MPDPC.

Simulation results
In order to prove the effectiveness of the proposed lowcomputation MPDPC for TL-FL-APF, the simulation results based on MATLAB/Simulink have been obtained using the parameters, as listed in Table 4.
First, in order to validate the limitation of reference zerosequence reactive power delay compensation, Figs. 11 and 12, respectively, represent the compensation effect considering three kinds of reference power delay compensation (RPDC) and only considering reference active and reactive power delay compensation (RARPDC).
As shown in Figs. 11a and 12a, both RPDC and RARPDC can realize the balance of three-phase grid currents. The THD of RPDC shown in Fig. 11b is 3.29%, and the THD of RARPDC shown in Fig. 12b is 2.30%.The THD of grid current decreases from 3.29 to 2.30%, indicating that ignoring the reference zero-sequence reactive power delay compensation is necessary. The neutral-wire current of two kinds of methods is shown in Figs. 11c and 12c. The neutral-wire current of RPDC contains two larger spikes in each fundamental cycle, and this is detrimental to the power system. The spikes are caused by defect of zero-sequence reactive power model and improper extrapolation of reference zerosequence reactive power. Figures 11d and 12d show the neutral-point potential fluctuation of TL-FL-APF. Through The power analysis results of RARPDPC covering p, q, and q 0 are shown in Fig. 13. p*, q*, and q* 0 are calculated from load current and grid voltage, and p c , q c , and q 0c are correspond to compensation power of TL-FL-APF. As it can be seen from Fig. 13, three kinds of compensation powers all follow the reference power well.
Then, low-computation MPDPC considers the reference delay compensation method proposed in this article. Figure 14 shows the information statistics of M, R in the low-computation MPDPC from 0.1 s to 0.101 s. When M is equal to ± 1 or ± 2, there are 18 vectors involved in voltage  Table 1. This effectively verifies the rationality and feasibility of low-computation MPDPC based on the method of spatial stratification. Figure 15 shows the compensation effect of lowcomputation MPDPC. As shown in Fig. 15a, b, the grid currents can be basically balanced and THD of grid current is  Figure 15c shows that neutral-wire current can be kept within 6A and has no current spikes. Figure 15d shows that neutral-point potential fluctuation is stable within 5.4 V. The power analysis results of p, q, and q 0 are shown in Fig. 16. Three kinds of compensation power also keep tracking of their reference value.
For further dynamic response investigation of APF, load changes from 10 to 7.5 in 0.3 s, and the results for two algorithms are depicted in Figs. 17 and 18. Its shows that the dynamic response of the grid current of both algorithms is approximate to 20 ms and has no current overshoot. Two algorithms have appropriate dynamic response under load changes.
In these two algorithms, the compensation ability of TL-FL-APF can be fully exerted, but the compensation effect is different. Classical MPDPC requires very huge operations to determine optimality (81 power predictions, 81 neutral-point voltage predictions, and 81 cost function evaluations).
As for low-computation MPDPC, multiple power prediction is converted into single voltage prediction through the principle of deadbeat control, and the vector range is reduced from 81 to 4-18, which greatly reduces the controller complexity. At the same time of power following, the control

Experimental results
To further verify the correctness of the proposed lowcomputation MPDPC strategy, the experiments based on control-hardware-in-loop are carried out. The hardware platform of experiments is shown in Fig. 19. The model of TL-FL-APF is developed in Typhoon402. Controller of TL-FL-APF is implemented in a TMS320F28335 DSP + FPGA control board, and the sampling frequency is set at 20 kHz. The parameters of experiment are the same as the simulations, which are listed in Table 4.  Figures 20a and 21a show that three-phase grid currents are balanced and no distortion. The neutralwire current is controlled around 0 V. In Figs. 20b and 21b, the capacitor voltages are kept balanced for two algorithms. Since redundant vector selection replaces 81 neutral-point potential prediction budgets, the neutral-point potential fluctuation is increased from 2 to 6 V. The power following for two algorithms is shown in Figs. 20c and 21c, and since the output of the voltage loop is superimposed on the reference active power, the active power following does not reflect the situation when APF does not work. It can be seen that compensation powers for two algorithms can keep tracking of reference powers.
Based on PQ 3198 analyzer, the THD analysis has been carried out from the experimental data of current waveforms for the two algorithms in Fig. 22. It can be observed that the proposed low-computation MPDPC improves gird current quality with a THD of 3.11%, compared with the PDPC proposed in [34], which provides a THD of 4.4%. The THD of experiments is basically consistent with simulation results.
The outstanding advantage of MPC is the fast dynamic response, and the dynamic responses with classical MPDPC and low-computation MPDPC are shown in Figs. 23 and 24. Its shows that the dynamic response of the grid current of both algorithms is approximate to 20 ms and has no current overshoot. As shown in Figs. 23b and 24b, APF compensation powers for two algorithms can keep tracking of reference powers under load switching. Therefore, low-computation MPDPC has the same fast dynamic response as classical MPDPC.
Based on the time slot monitor of Typhoon HIL 402, the RTED of two algorithms is displayed in Fig. 25. The results  of the experiment can be sorted out as given in Table 5. The RTED of classical MPDPC is found to be 42 μs, which is 31.6% more than low-computation MPDPC. The max RTED of controller has been reduced to 26.2 μs compared with the low-computation strategy proposed in [28], which provides a RTED of 21.8 μs.

Conclusion
Aiming at the excessive prediction computation for TL-FL-APF, this article proposed a low-computation MPDPC. Based on generalized instantaneous reactive power theory, a new predictive power model is established, and a reference delay compensation method is given. The principle of deadbeat control and spatial stratification are used to convert 81 times power predictions into single voltage prediction and reduce optimal vector range from 81 to 4-18. Simulation and experimental results are provided to demonstrate lowcomputation MPDPC. The low-computation MPDPC leads to approximately 31.6% reduction in the RTED of classical MPDPC. In addition, its dynamic response is almost as fast as classical MPDPC. However, since the influence of nonredundant vector is not considered, the neutral-point potential still has some offset. The following work is to decline neutralpoint potential offset to improve compensation accuracy.