Comprehensive Predictive Control Model for a Three-Phase Four-Legged Inverter

Abstract

This paper presents a comprehensive model predictive control (CMPC) method to control a three-phase four-legged inverter (TP4LI) for PV systems. The proposed TP4LI model aims to predictively model and control switching frequency and higher voltage/current switching to reduce losses. The CMPC model can be operated in four modes, namely standard MPC mode (Mode I), switching frequency reduction (SFR) mode (Mode II), high voltage/current switching loss reduction (SLR) mode (Mode III), and SFR plus SLR mode (Mode IV, a combination of Modes II and III). The proposed CMPC approach controls the TP4LI to (1) successfully track balanced and unbalanced reference currents with balanced or unbalanced loads; (2) reduce switching losses; and (3) keep the generated current total harmonic distortion (THD) within the industry’s recommended limits. The TP4LI model with the CMPC approach was verified and validated in the MATLAB/Simulink for a PV system. The simulation results show good tracking and performance of the TP4LI for balanced and unbalanced reference currents with balanced and unbalanced loads in all four modes of operation.

1. Introduction

Inverters are used to connect renewable energy systems (wind, PV, etc.) to a grid [1,2] since they convert DC power into AC power for loading and/or injection into the grid. Inverters can be single-, two-, or three-phase, with three-phase inverters being three- or four-legged. Three-legged inverters are used for balanced three-phase loads, while four-legged inverters are used for balanced three-phase as well as unbalanced three-phase loads and injection into the grid. Three-phase three-legged inverter has a total of 8 possible switching states (2³ = 8), while a three-phase four-legged inverter (TP4LI) has 16 (2⁴ = 16) possible switching states due to an additional fourth leg. The fourth leg adds a neutral current-carrying capability to the inverter, extending the inverter’s use for unbalanced load in addition to balanced load.

The literature has a rich repository on all kinds of inverter control techniques. A review of control techniques for single-phase inverters is presented by [3,4], and a review of control techniques for three-phase inverters is presented by [5,6]. Repetitive control is reported by [7,8], sliding mode is reported by [9,10], dead-beat control is reported by [11,12], synchronous reference frame is reported by [13], sine pulse-width modulation is reported by [14,15], standard space-vector pulse-width modulation is reported by [16,17], and high-current, not-switched space-vector pulse-width modulation is reported by [18]. However, the studies by [7,8] suffer from poor tracking accuracy and large memory requirement with poor non-periodic disturbance rejection ability [13]. Sliding mode [9,10] and dead-beat control [11,12] are complex and sensitive to parameter variations and suffer from varying loading conditions. PI regulators used in synchronous reference frame [13] controls require regulator adjustments and have poor disturbance rejection capability. Sine pulse-width modulation [14,15] control needs a carrier and modulating wave generation and cannot exploit the full potential of the DC source. Space-vector pulse-width modulation [16,17,18] is complex and time-consuming due to exhaustive vector calculations, and it is non-intuitive for software/hardware implementation [19,20].

Model predictive control (MPC) is simple and intuitive and eliminates modulation stages and linear regulators [21,22,23]. MPC flexibility allows the incorporation of a system’s nonlinearities and limitations into the model, resulting in a better steady state and faster transient response when compared to traditional controllers [24,25,26]. MPC used for a multitude of applications in renewable energy systems is reported in the literature. The use of MPC for energy dispatch problems in electrical energy systems with many intermittent renewable resources is reported by [27]. The use of MPC for economic optimization of PV plants with energy storage is presented by [28]. The use of MPC for resource coordination in renewable energy-based microgrids is reported by [29]. Recently, MPC is used for inverter control in a system’s model to predict and control the future behavior of the inverter [30,31], using both present and past data.

A review of MPC application in power converters is presented by Ref. [32]. MPC of a three-phase inverter is compared against hysteresis and pulse-width modulation control techniques in [31]. MPC’s superiority in terms of harmonics is reported by [33]. A unified DC and AC side control for PV systems using MPC by predicting the output current based on the PV potential of the PV array on the DC side is presented by [34]. MPC of a utility-interactive single-phase inverter is reported by [35] and of a utility-interactive seven-level single-phase inverter by [36]. The control of active and reactive power using MPC is reported by [37]. Low switching frequency MPC of a three-phase module integrated with DC voltage boost capability and control of active and reactive power with low voltage ride through capability is examined by [38]. However, none of the above studies has investigated, modeled, and reported switching frequency (SFR) and higher voltage–current switching loss reduction (SLR) based on a comprehensive MPC for a TP4LI.

Ref. [39] proposed a model-predictive direct power control strategy for flexible power regulation of a grid-tied inverter that is used in PV systems with a switching frequency reduction algorithm. However, this study did not focus on high voltage–current switching to reduce losses. Its focus was a three-legged inverter, which is good for balanced loads only. Refs. [20,21,40] proposed an MPC strategy for a two-level four-legged inverter to deliver power to grids with balanced and unbalanced loads but without high voltage–current state-switching loss reduction. Ref. [41] proposed a diode rectifier-based three-level boost converter with a neutral-point-clamped inverter. However, switching frequency reduction is limited to the neutral leg only for the neutral-point-clamped inverter. Additionally, high voltage–current state-switching loss reduction is not the focus. Ref. [42] proposed an MPC for optimal switching reduction calculations by preselecting five possible switching combinations, and [43] proposed a current-limiting MPC in which any switching combination producing higher than the desired current is excluded from the list of prospective switching options in the next interval. Although successful in achieving their goals, [42,43] did not report switching frequency and high voltage–current switching loss reduction based on an MPC for a TP4LI.

This paper proposes switching frequency reduction (SFR) and high voltage–current switching loss reduction (SLR) using a comprehensive MPC (CMPC) for a TP4LI. The CMPC is used to control the TP4LI in four modes, namely (1) MPC-only mode (Mode I), (2) SFR mode (Mode II), (3) SLR mode (Mode III), and (4) SFR plus SLR mode (Mode IV) with balanced and unbalanced reference currents and loads. The rest of the paper is organized as follows: (Section 1) introduction of the TP4LI model; (Section 2) materials and methods; (Section 3) results and discussions; and (Section 4) conclusions.

The novelty and main objectives of the proposed CMPC, besides the inherent MPC advantages, are to provide a simple yet intuitive control model for a TP4LI that is (A) decoupled to independently minimize losses associated with (1) switching frequency or (2) high voltage–current switching and yet (B) integrative to minimize losses associated both with A(1) and A(2) simultaneously, while (C) keeping the harmonics under industry standard limits [44,45]. The proposed method is an MPC, which inherently draws on the strengths of MPC and, thus, boasts small memory requirement, computational speed, simplicity, parameter robustness, good performance, and no carrier/modulating wave generation requirements, as discussed in the second paragraph of this section.

2. Materials and Methods

The model predictive control (MPC) strategy takes advantage of the fact that only a finite number of possible switching states are associated with a three-phase four-legged inverter (TP4LI). These states are discrete, and the model of the TP4LI can be used in association with a discrete-time model of the load to predict the behavior of the system over a prediction horizon (sampling time). The objective of the MPC scheme is to predict the switching state and, thus, the current by tracking the reference (or desired) current with minimum error. For each sampling period, the resulting (predicted) output current ($i_o^p$) is measured and compared with the reference current (${\mathrm{i}}_{\mathrm{o}}^{*}$) to minimize error. The sampling period (T_S) is selected to be small such that the reference current does not change significantly during that period (T_S).

Figure 1 shows a TP4LI circuit with a filter inductor (L) and a load resistor (R). The TP4LI has 4 legs with 8 switches. The fourth leg, also known as the neutral leg, enables the independent generation of voltage and current on each of the three phases of the TP4LI. The TP4LI’s 8 switches result in 16 (2⁴) possible switching states, as tabulated in

Table 1. LI Switching States with Corresponding Voltages and Vectors.

Switches				Voltage			Vector
SA	SB	SC	Sn	VAn	VBn	VCn	Vx
n	n	n	n	0	0	0	V0
n	n	n	p	−V	−V	−V	V1
n	n	p	n	0	0	V	V2
n	n	p	p	−V	−V	0	V3
n	p	n	n	0	V	0	V4
n	p	n	p	−V	0	−V	V5
n	p	p	n	0	V	V	V6
n	p	p	p	−V	0	0	V7
p	n	n	n	V	0	0	V8
p	n	n	p	0	−V	−V	V9
p	n	p	n	V	0	V	V10
p	n	p	p	0	−V	0	V11
p	p	n	n	V	V	0	V12
p	p	n	p	0	0	−V	V13
p	p	p	n	V	V	V	V14
p	p	p	p	0	0	0	V15

, with corresponding A-, B-, and C-phase voltages and vectors ranging from 0 to 15. The switches are switched in a controlled manner to connect the A, B, and C terminals to the positive (P) or negative (N) terminals of the DC source (V_DC) to generate v and i.

The output voltages of the TP4LI are calculated based on Equations (1)–(3):

$${\mathrm{V}}_{\mathrm{An}}=\left({\mathrm{S}}_{\mathrm{A}}-{\mathrm{S}}_{\mathrm{n}}\right){\mathrm{V}}_{\mathrm{DC}}$$

(1)

$${\mathrm{V}}_{\mathrm{Bn}}=\left({\mathrm{S}}_{\mathrm{B}}-{\mathrm{S}}_{\mathrm{n}}\right){\mathrm{V}}_{\mathrm{DC}}$$

(2)

$${\mathrm{V}}_{\mathrm{Cn}}=\left({\mathrm{S}}_{\mathrm{C}}-{\mathrm{S}}_{\mathrm{n}}\right){\mathrm{V}}_{\mathrm{DC}}$$

(3)

Similarly, load dynamics are modeled by (4), where L and R are the filter inductance and resistance, i is the current, and v is the TP4LI-generated voltage vector. The current i in (4) can be approximated by (5) using the Euler forward equation, where k is the present interval and k + 1 is the next interval for which the prediction is made.

$$\mathrm{v}=\mathrm{L}\frac{\mathrm{di}}{\mathrm{dt}}+\mathrm{Ri}$$

(4)

$$\frac{\mathrm{di}}{\mathrm{dt}}\cong \frac{\mathrm{i}\left(\mathrm{k}+1\right)-\mathrm{i}\left(\mathrm{k}\right)}{\mathrm{T}}$$

(5)

Equation (6) is the result of (4) and (5), which is further expressed by (7) to exercise the MPC in a αβ reference frame and predict the load current for each switching state of the TP4LI model.

$${\mathrm{i}}^{\mathrm{p}} \left(\mathrm{k}+1\right)=\left(1-\frac{\mathrm{RT}}{\mathrm{L}}\right)\mathrm{i}\left(\mathrm{k}\right)+\frac{\mathrm{T}}{\mathrm{L}}\mathrm{v}\left(\mathrm{k}\right)$$

(6)

$$\left[\begin{array}{c}{\mathrm{i}}_{\mathsf{\alpha }\left(\mathrm{k}+1\right)}\\ {\mathrm{i}}_{\mathsf{\beta }\left(\mathrm{k}+1\right)}\end{array}\right]=\left[\begin{array}{cc}1-\frac{\mathrm{RT}}{\mathrm{L}}& 0\\ 0& 1-\frac{\mathrm{RT}}{\mathrm{L}}\end{array}\right]\left[\begin{array}{c}{\mathrm{i}}_{\mathsf{\alpha }\left(\mathrm{k}\right)}\\ {\mathrm{i}}_{\mathsf{\beta }\left(\mathrm{k}\right)}\end{array}\right]+\left[\begin{array}{cc} \frac{\mathrm{T}}{\mathrm{L}}& 0\\ 0& \frac{\mathrm{T}}{\mathrm{L}}\end{array}\right]\left[\begin{array}{c}{\mathrm{v}}_{\mathsf{\alpha }\left(\mathrm{k}\right)}\\ {\mathrm{v}}_{\mathsf{\beta }\left(\mathrm{k}\right)}\end{array}\right]$$

(7)

It is evident from (6) (or (7)) that the current prediction requires a definition of the loads v and i. However, defining or changing the switch state from on to off or from off to on state to define v and i causes power loss, and frequent changes in switching states increase these power losses. Hence, switching frequency reduction can improve the TP4LI’s efficiency.

Starting at V₀ = 0000 (S_AS_BS_CS_n sequence) as a present state, there is a total of 15 possible switching paths (V₁ to V₁₅), as shown in Table 1. Four switching paths have 1 switch-state change (V₁, V₂, V₄, and V₈), 6 switching paths have 2 switch-state changes (V₃, V₅, V₆, V₉, V₁₀, and V₁₂), 4 switching paths have 3 switch-state changes (V₇, V₁₁, V₁₃, and V₁₄), and 1 switching path has 4 switch-state changes (V₁₅), as shown in Figure 2. This is true for switching paths from any switch-state as a starting status. Thus, power loss reduction is possible if control is implemented such that the minimum number of switches change their states at any time, providing that the TP4LI’s performance is not degraded significantly and the THD is kept within the industry standard limits.

Switching losses also depend on the magnitudes of the voltage across and the current through the switch at the time of state switching, besides switching frequency. Therefore, the state change of a switch with a higher voltage and a higher current causes a higher power loss compared to the state change of a switch with a lower voltage and a lower current. Hence, switching to a switch state with a lower voltage and current is preferred to reduce further power losses associated with the switches, providing the TP4LI’s performance is not affected and the THD is kept within the industry’s standard limits.

This work proposes the following three objectives to achieve the objectives outlined above:

(1)

J₁: Minimize the error between the predicted and reference currents as follows:

$${\mathrm{J}}_1={({\mathrm{i}}_{\mathsf{\alpha }}^{*}-{\mathrm{i}}_{\mathsf{\alpha }}^{\mathrm{p}})}^2+{({\mathrm{i}}_{\mathsf{\beta }}^{*}-{\mathrm{i}}_{\mathsf{\beta }}^{\mathrm{p}})}^2$$

(8)

where ${\mathrm{i}}_{\mathsf{\alpha }}^{\mathrm{p}}$ and ${\mathrm{i}}_{\mathsf{\beta }}^{\mathrm{p}}$ are the predicted currents, and ${\mathrm{i}}_{\mathsf{\alpha }}^{*}$ and ${\mathrm{i}}_{\mathsf{\beta }}^{*}$ are the reference currents.

(2)

J₂: Minimize the state-change switching frequency of switches:

$${\mathrm{J}}_2={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{|\mathsf{\lambda }}_{\mathrm{SFR}} ({\mathrm{S}}_{\mathrm{k}}-{\mathrm{S}}_{\mathrm{k}-1})|$$

(9)

where λ_SFR is the switching frequency reduction weighting factor, and S_k and S_k−1 denote the present and the previous interval switch states for the TP4LI. This objective reduces the number of switch-state changes between previously applied and presently synthesized switching combinations, i.e., it optimizes the number of switch-state changes between the intervals with minimal error. Higher values of λ_SFR may produce distorted currents; therefore, the selection of this weighting factor requires careful attention.

(3)

J₃: Change the state of a switch with lower voltage and current with higher priority:

$${\mathrm{J}}_3=\left|{\mathsf{\lambda }}_{\mathrm{SLR}} \left({\mathrm{V}}_{\mathrm{k}}{\mathrm{I}}_{\mathrm{k}}-{ \mathrm{V}}_{\mathrm{k}-1}{\mathrm{I}}_{\mathrm{k}-1}\right)\right|$$

(10)

where λ_SLR is the weighting factor for switching losses. The voltage and current in the current interval and previous interval are represented by V_k, I_k, and V_k−1, I_k−1, respectively. This objective reduces switching losses associated with higher voltage/current switch-state changes. Like λ_SFR, higher values of λ_SLR may also lead to distorted currents.

However, these objectives should be achieved while maintaining the performance and the THD of the TP4LI within industry limits, i.e., THD should be less than 5% as per the IEEE 519 standard. These conditions are introduced as constraints in the MPC model. It should be noted that, if the sampling period is small, we do not require extrapolation [20,42,43].

Combining (8)–(10), we obtain the objective function for the proposed MPC method as follows:

$$\mathrm{J}={\mathrm{J}}_1+{\mathrm{J}}_2+{\mathrm{J}}_3$$

(11)

Based on the two tuning parameters, λ_SFR (switching frequency reduction weighting factor) and λ_SLR (weighting factor for switching losses), we define four modes of operation for the proposed controller:

• Mode I: λ_SFR = λ_SLR = 0

Standard MPC operation ignoring J₂ and J₃. This achieves a reduction in the issues associated with conventional techniques only, as outlined in the 2nd paragraph of Section 1 and as identified by the references cited in the same section.

• Mode II: λ_SFR ≠ 0 and λ_SLR = 0

SFR-based MPC operation ignoring J₃. In addition to the benefits of Mode I, this mode achieves switching frequency reduction and, thus, minimizes losses due to excessive and unnecessary switching associated with most of the conventional approaches.

• Mode III: λ_SFR = 0 and λ_SLR ≠ 0

SLR-based MPC operation ignoring J₂. In addition to the benefits of Mode I, this mode achieves high voltage–current switching reduction and, thus, minimizes losses due to switching the switches with a high voltage across and a high current through them, which is again a factor that causes significant losses in most of the conventional approaches.

• Mode IV: λ_SFR ≠ 0 and λ_SLR ≠ 0

SFR plus SLR-based MPC operation activating all objectives. This mode combines the benefits of Mode I, Mode II, and Mode III and, thus, boasts (1) a reduction in higher frequency switching losses, (2) a reduction in high voltage–current switching losses, and (3) a reduction in the issues listed in Section 1 that are associated with conventional controllers, while keeping the THD within the industry limits (also attained in the other modes).

The objective function J (11) is evaluated for each of the 16 possible voltage vectors (V₀ to V₁₅) generated by the TP4LI. The optimal J with the minimum error between the predicted (present) and measured (past) outputs is selected, and the corresponding control action (switching combination) from a set of control actions (0000 to 1111) is identified to be applied in the next interval (sampling period). The output produced is observed, and the process is repeated to synthesize the optimal control action for the next interval.

3. Results and Discussions

To evaluate the effectiveness of the proposed comprehensive MPC (CMPC) strategy for the TP4LI, we considered a system configuration that consists of a DC source (a PV system), an AC grid, and a 4-wire load. The proposed controller receives the output from the TP4LI and gives the control signal to operate the TP4LI.

The model was simulated multiple times to test the effectiveness of the proposed CMPC model in all four modes of operation: standard MPC (Mode I), SFR (Mode II), SLR (Mode III), and SFR + SLR (Mode IV) with balanced reference currents and balanced loads. Additionally, Mode I for unbalanced reference current tracking and unbalanced loads was also simulated with successful current tracking, although the results are not discussed further in this work. Only selected results are reported to avoid increasing the length of the paper. The parameters used for the simulations are listed in

Table 2. Simulation Parameters.

Parameters	Values
R	5 Ω
L	10 m H
VLn	277 V
f	60 Hz
Ts	1 μ sec
VDC	≥√2 VLn V
I	10 A
Simulation Time	0.075 s

(Simulation Parameters). Refs. [46,47,48] were used for switching loss estimations, and readers interested in further readings are referred to these studies for details on switching loss estimation. The weighting factors, λ_SFR and λ_SLR, were selected to reduce the losses associated with switching frequency and higher voltage–current switching, while keeping the THD within the industry standard limits.

Figure 3 shows the responses of the TP4LI in Mode I (MPC with λ_SFR = λ_SLR = 0, i.e., standard MPC operation ignoring both J₂ and J₃) with balanced reference currents. Figure 3 displays the 3-phase voltages, the 3-phase produced currents with 3-phase reference currents, the neutral current, the switching frequency (one duty cycle), and the switching losses for the four legs.

Figure 3a shows that voltage V_abc is perfectly maintained, leading to 3-phase currents (I_abc), which successfully track the balanced reference currents, as shown in Figure 3b, with neutral current (I_n) maintained at zero, as shown in Figure 3c. The total harmonic distortion (THD) for phase A current (I_a-THD) is 0.06% for this mode.

Moreover, switch S₁ of phase A switched 13,488 times (as reported by the switch flipping counter in the model) for the duration of the simulation, with Figure 3d showing the switching frequency (f_sw) in one duty cycle. Finally, Figure 3e shows the switching power losses (P_sw). The total loss is 431.6 W, with a loss of 08.04 W, 113.77 W, 92.22 W, and 117.61 W for Leg₁, Leg₂, Leg₃, and Leg_n, respectively, for the duration of the simulation, as shown in Figure 3e.

Figure 4 shows the responses of the TP4LI in Mode II (MPC with λ_SFR = 0.015 and λ_SLR = 0, i.e., SFR-based MPC operation ignoring J₃) with balanced reference currents. Figure 4 displays the 3-phase voltages, the 3-phase produced currents with 3-phase reference currents, the neutral current, the switching frequency (one duty cycle), and the switching losses for the four legs.

Figure 4a shows that voltage V_abc is perfectly maintained, leading to 3-phase currents (I_abc), which successfully track the balanced reference currents, as shown in Figure 4b, but with increased THD and neutral current (I_n) greater than zero, as shown in Figure 4c. The total harmonic distortion (THD) measured is 2.11% for phase A current (I_a-THD) for this mode.

However, switch S₁ of phase A switched only 3159 times for the duration of the simulation, with Figure 4d showing the switching frequency (f_sw) for one duty cycle. Finally, Figure 4e shows the switching power losses (P_sw). The loss also decreases to a total of 101.19 W, with a loss of 25.31 W, 24.45 W, 25.63 W, and 25.08 W for Leg₁, Leg₂, Leg₃, and Leg_n, respectively, for the duration of the simulation, as shown in Figure 4e.

Figure 5 shows the responses of the TP4LI in Mode III (MPC with λ_SFR = 0 and λ_SLR = 0.015, i.e., SLR-based MPC operation ignoring J₂) with balanced reference currents. Figure 5 displays the 3-phase voltages, the 3-phase produced currents with 3-phase reference currents, the neutral current, the switching frequency (one duty cycle), and the switching losses for the four legs.

Figure 5a shows that voltage V_abc is perfectly maintained, leading to 3-phase currents (I_abc), which successfully track the balanced reference currents, as shown in Figure 5b. The THD for this mode of operation is lower than Mode II but higher than Mode I. Similarly, the neutral current (I_n) is greater than zero and higher than Mode I but lower than Mode II, as shown in Figure 5c. The total harmonic distortion (THD) measured is 0.89% for phase A current (I_a-THD) for this mode.

Following the trend, switch S₁ of phase A switched only 9195 times (higher than Mode II but lower than Mode I) for the duration of the simulation, with Figure 5d showing the switching frequency (f_sw) for one duty cycle. Finally, Figure 5e shows the switching power losses (P_sw). The total loss is 345.98 W, with a loss of 73.65 W, 103.53 W, 66.38 W, and 102.42 W for Leg₁, Leg₂, Leg₃, and Leg_n, respectively, for the duration of the simulation, as shown in Figure 5e.

Note that f_sw and P_sw are lower than Mode I but higher than Mode II for this mode (Mode III), but the THD is higher than Mode I and lower than Mode II with the same weighting factor of 0.015 (λ_SFR = λ_SLR = 0.015). Thus, λ_SLR has a shallower influence compared to λ_SFR. I_n is also noticed to have a smaller change when compared to Mode I and Mode II for λ_SFR = λ_SLR = 0.015.

Figure 6 shows the responses of the TP4LI in Mode IV (MPC with λ_SFR = 0.015 and λ_SLR = 0.025, i.e., SFR+SLR-based MPC operation including both J₂ and J₃) with balanced reference currents. Figure 6 displays the 3-phase voltages, the 3-phase produced currents with 3-phase reference currents, the neutral current, the switching frequency (one duty cycle), and the switching losses for the four legs.

Figure 6a shows that voltage V_abc is perfectly maintained, leading to 3-phase currents (I_abc), which successfully track the balanced reference currents, as shown in Figure 6b. THD for this mode of operation was measured at 1.84%, which is within the industry limits.

The number of times switch S₁ of phase A switched was observed to be 2090, with Figure 6d showing the switching frequency (f_sw) for one duty cycle. Finally, the switching power losses (P_sw) decrease to a total loss of 77.39 W, with a loss of 16.74 W, 19.10 W, 14.43 W, and 27.12 W for Leg₁, Leg₂, Leg₃, and Leg_n, respectively, for the duration of the simulations as shown in Figure 6e.

A comparison of Mode IV to Mode I, Mode II, and Mode III is presented in

Table 3. Comparison of the four modes of operation.

Performance Metric	Mode I	Mode II	Mode III	Mode IV
THD for current (%)	0.06	2.11	0.89	1.84
Number of switch flips	13,488	3159	9195	2090
Power loss	431.6	101.19	345.98	77.39

. The table clearly shows that Mode IV outperforms the other three modes, with significant performance improvements. Mode IV shows a significant reduction both in terms of switching frequency (number of times a switch flips) and power loss, while keeping the THD within the industry limits.

Table 4. THD as a function of λ_SFR and λ_SLR.

THD @ λSFR		THD @ λSLR
λSFR	THD (%)	λSLR	THD (%)
0.01	1.44	0.01	0.65
0.025	3.55	0.025	1.35
0.05	7.3	0.05	4
0.1	14.79	0.1	8.54
0.25	38.23	0.25	23.55
0.5	75.19	0.5	45.12
1	155.49	1	50.61

tabulates I_a-THD as a function of λ_SFR and λ_SLR from 0.01 to 1. The table is also reproduced as a graph in Figure 7. Both the table and the figure confirm what was observed during the simulations, that is, λ_SFR has a stronger influence on the THD when compared to λ_SLR. Thus, a larger value of λ_SLR is required to reduce the P_sw losses compared to λ_SLR.

4. Conclusions

This paper proposed a comprehensive MPC (CMPC) method to reduce losses associated with switching frequency and higher voltage–current switching for a three-phase four-legged inverter (TP4LI). The proposed CMPC method is simple yet intuitive and provides (A) decoupled control for the TP4LI to independently minimize losses associated with (1) switching frequency or (2) high voltage–current switching and (B) integrative control of both A(1) and A(2), while (C) keeping the harmonics under the industry standard limits. The proposed CMPC method is based on MPC; therefore, it uses small memory, offers fast computational speed and parameter robustness, requires no carrier/modulating wave generation, and has good performance.

The TP4LI was operated and controlled using the proposed CMPC-based controller in four different modes: standard MPC (Mode I), switching frequency reduction (Mode II), switching loss reduction (Mode III), and switching frequency plus switching loss reduction (Mode IV) for balanced reference current tracking. Additionally, Mode I (standard MPC) was simulated for unbalanced reference current tracking and unbalanced loads. The results showed that (1) balanced and unbalanced reference currents were successfully tracked for balanced and unbalanced loads; (2) switching frequency losses were reduced in Mode II; (3) switching losses associated with higher voltage–current switch flipping were reduced in Mode III; and (4) both switching frequency and high current/voltage switching losses were reduced in Mode IV.

The simulation results showed that (1) balanced and unbalanced reference currents were successfully tracked for balanced and unbalanced loads; (2) switching frequency losses were reduced in Mode II; (3) losses associated with higher voltage/current switching were reduced in Mode III; and (4) both switching frequency and high current/voltage switching losses were reduced in Mode IV. The proposed controller enables decoupled control in terms of switching frequency or high voltage–current; however, Mode IV has superior performance both in terms of reduction in switching and losses. The THD values for currents were observed to be below the industry standard limits during all four modes of operation.

The authors see the proposed CMPC’s suitability for inverter control in all DC–AC conversion applications, especially in the ever-rising renewable energy sector, such as PV and wind. The authors plan to conduct experimental work to further this work in the future, especially in comparison with the sine PWM and space-vector PWM techniques. Additionally, a smaller-scale experimental setup in a laboratory with a PV module (on the roof), a charge controller, a battery, and an inverter controlled by the CMPC connected to a grid is also in the planning to further this work.