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An Improved Sequential-Model Predictive Control for an O-Z-Source Inverter Without Weighting Factors
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Xuan WANG, Wei LUO
CPSS Transactions on Power Electronics and Applications | 2024, 9(2) : 207 - 218
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CPSS Transactions on Power Electronics and Applications | 2024, 9(2): 207-218
• Papers •
An Improved Sequential-Model Predictive Control for an O-Z-Source Inverter Without Weighting Factors
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Xuan WANG, Wei LUO
Affiliations
  • University of Shanghai for Science and Technology Department of Electrical Engineering Shanghai China

    • Xuan Wang was born in Shandong, China, in 1997. She received the B.S. degree in Electronic Information Engineering from Shanghai University of Technology, China, in 2020. She is currently pursuing the Master degree in electric engineering with University of Shanghai for Science and Technology, Shanghai, China. Her current research interests include Z-source inverter.

    Wei Luo was born in Shanxi, China, in 1986. He received the B.S. degree in electrical engineering and automation from the China University of Mining and Technology, Xuzhou, China, in 2009, M.S. and Ph.D. degrees in electric engineering from Shanghai Jiao Tong University, Shanghai, China, in 2012 and 2018. He is currently an Associate Professor of electrical engineering with University of Shanghai for Science and Technology, China. His research interests include multilevel converters, high-power motor drives, and internet of things.

Published: 2024-06-10 doi: 10.24295/CPSSTPEA.2024.00002
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When finite set model predictive control is applied to an OZsource inverter (OZSI) containing a transformer, there are multiple control variables, it is difficult to adjust the weighting factors, and the currents on both sides of the transformer could change abruptly, making it impossible to calculate and derive reference values to directly predict and control the currents on both sides of the transformer. In this paper, an improved sequentialmodel predictive control is proposed for OZSI with a transformer without adjusting the weighting factors. By equating the transformer as a parallel connection of the excitation inductance with a set of ideal transformers without adjusting the weighting factors, the reference value of the magnetization current of the excitation inductance can be calculated according to the theoretical derivation, which can realize the predictive control of the OZSI. Simulation analysis and experimental results show that the proposed control method achieves the sequentialmodel predictive control of OZSI without adjusting the weight factors, with good steadystate and dynamic characteristics.

O-Z-source inverter  /  sequential-model prediction  /  transformer  /  weighting factors.
Xuan WANG, Wei LUO. An Improved Sequential-Model Predictive Control for an O-Z-Source Inverter Without Weighting Factors[J]. CPSS Transactions on Power Electronics and Applications, 2024 , 9 (2) : 207 -218 . DOI: 10.24295/CPSSTPEA.2024.00002
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I. Introduction
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THE Z-source inverter (ZSI) could achieve a boost function by the shoot-through state, however the ZSI has a large start-up inrush current, and in the impedance network, the stress on the capacitor voltage and the inductor current is significant [1],[2]. The quasi-Z-source inverter (q-ZSI) is an improvement of the ZSI. Not only does it inherit the advantages of the ZSI, but it also has the benefits of reduced stress on the capacitor voltage and inductor current, as well as continuous input current. However, the output voltage gain has not been increased [3].[4],[5] proposed transformer-based Z-source networks, Trans-Z-source converters and $\Gamma$-Z-source converters, effectively enhance power density, reduce system costs, and improve the system’s boosting capability through the precise control of the transformer’s turns ratio.
However, they still have the problem of discontinuous power input current and starting inrush current [6],[7]. Without increasing the system cost, the O-Z source inverter based on O-type impedance network applied in this paper can reduce the capacitor voltage stress and effectively suppress the start-up inrush current. The O-Z source topology is similar to the Trans-Z source and $\Gamma -\mathrm{Z}$ source, both of which can be classified as a transformer-based Z source topology. Compared to existing Z-source topologies, the boosting capacity can be improved by adjusting the transformer turn ratio, effectively solving the excessive stress on the capacitor, and suppressing the start-up inrush current with fewer required components and a compact structure [8]-[10]. Based on the above advantages, the O-ZSI can be applied to the grid-connected renewable energy generation systems [11].
In recent years, with the rapid development of digital signal processing technology, finite-control-set MPC (FCS-MPC) has many advantages such as a simple control principle, easy digital implementation and fast dynamic response [12]-[14]. Many scholars have applied FCS-MPC to the control of Z-source topology and achieved good performances [15]-[17]. However, FCS-MPC in the above literature is applied to Z-source topologies without transformers (ZSI or q-ZSI, etc.). For Z-source topologies containing transformers (trans-ZSI, $\Gamma$-ZSI or O-ZSI), the currents on both sides of the transformer will abruptly change [18]. Unlike the reference value of the inductor current can be calculated according to theoretical derivation in the inductor current ZSI or q-ZSI [19]-[21], so it is not possible to directly predict and control the current on both sides of the transformer. In this paper, the O-ZSI is equivalent to the parallel connection of a set of ideal transformers of magnetizing inductance with a set of ideal transformers. The reference value of the magnetization current ${i}_{\mathrm{m}}$ of the magnetizing inductance is derived according to the theory, and the transformer is controlled by controlling the magnetization current. Then, the FCS-MPC of O-ZSI with transformer is realized.
In the O-ZSI control system, the cost function constructed by the conventional FCS-MPC usually includes three control objects: inductor current, capacitor voltage, and output current [22]-[24], each of which should have a weighting factor to reflect the importance of the control quantity [25]-[27]. However, weighting factors are often selected based on experiences and lack theoretical guidance [27]. Many scholars have examined how to eliminate the weighting factors. In the field of electric machines,[28] calculated the optimal weighting factors of stator flux linkage online based on the principle of minimum torque ripple. However, the mathematical expression of the optimal weighting factors is complex and depends on accurate motor parameters. In [29], an algebraic weighting factor selection method was proposed to achieve similar performance for predictive current control. Although the calculation of the weighting factors is simpler than that in [28], it still depends on the electrical parameters and system state. In [30], a multi-objective optimization method based on ranking is proposed. The cost function value of the torque and stator flux linkage in each switching state is transformed into a ranking value; then, the switching state with the smallest average ranking value is selected as the optimal switching state. Although this method avoids to design weighting factors, it increases the computational burden and control complexity. In [31], Professor Rodriguez proposed a sequential-model predictive control (S-MPC) strategy applied to the control of a motor by first selecting the two smallest switching states using the cost of torque and subsequently selecting the least costly switching state from two alternative switching states using the cost function of the magnetic flux, which eliminated the weighting factors of the traditional MPC in motor control field. However, this method has not been extended to the inverter with a transformer, and removing many switching states at one time may lead to inaccurate results.[32] analyzes the current by transforming it into the ${\alpha \beta }$ coordinate system, which introduces an additional control object and increases a cost function. This approach improves the accuracy of the results.[33],[34] apply this approach though improves the cost function and eliminates weighting factors by using the S-MPC method. However, the excessive elimination of voltage vectors each time may led to a reduction in prediction accuracy.
This paper applies a sequential-model predictive control to O-ZSI and proposes a sequential-MPC without weighting factors for O-ZSI. First, the O-ZSI topology is analyzed, its discrete mathematical model is established, and the prediction functions of the inductor current, capacitance voltage, and output current are derived. Then, the steady-state model of the O-ZSI is analyzed, the reference values of each control variable are derived, and the sequential-MPC method applied to the O-ZSI is proposed. Finally, the simulation analysis based on MATLAB/Simulink is performed, and an experimental prototype is built to verify the correctness and effectiveness of the proposed strategy in the O-ZSI with a transformer.
II. O-ZSI MODEL AND ANALYSIS
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A. Discrete Mathematical Model for O-ZSI
The topology of the O-ZSI is shown in Fig. 1 and consists of the power supply ${v}_{\text{in }}$, an O-Z source network, a three-phase inverter bridge, and a three-phase resistive inductive load. The O-Z source network consists of diode D, capacitor $C$ and a double-winding transformer, which is composed of the dotted terminals of the double-winding transformer, with the primary and secondary winding turn ratio $\gamma$.
$\gamma ={N}_{1}/{N}_{2}$
where ${N}_{1}$ is the number of turns of the primary winding of the double-winding transformer, and ${N}_{2}$ is the number of turns of the secondary winding of the double-winding transformer.
The O-ZSI and ZSI have similar basic principles and can be divided into two operating cases: the shoot-through state and the non-shoot-through state. The equivalent circuit for the two operating cases of O-ZSI is shown in Fig. 2. In the diagram, the transformer is equivalent to the parallel connection of the magnetizing inductance ${L}_{\mathrm{m}}$ and a set of ideal transformers.
Considering all possible switching states of the six switches, Table I lists the output voltage vectors of the inverter for the simplified eight switching states.
The output voltage vector can be expressed as follows:
${V}_{x}= \frac{2{U}_{\mathrm{{dc}}}}{3}\left({{S}_{\mathrm{A}}+ a{S}_{\mathrm{B}}+ {a}^{2}{S}_{\mathrm{C}}}\right)$
where the unit vector $a ={\mathrm{e}}^{\mathrm{j}{2\pi }/3}$ represents the ${120}^{\circ }$ phase difference between phases, $x =\left\lbrack {0,7}\right\rbrack .{U}_{\mathrm{{dc}}}$ is the DC-link voltage of the inverter after boosting by the O-Z network. SA, SB, and SC are the switching states of the three-phase upper arms of inverter bridges A, B, and C, respectively.
The current dynamic equation of the load current in the stationary ${\alpha \beta }$ coordinate system is:
$\left\{\begin{array}{l} L\frac{\mathrm{d}{i}_{\alpha }}{\mathrm{d}t}= {V}_{\alpha }- R{i}_{\alpha }\\ L\frac{\mathrm{d}{i}_{\beta }}{\mathrm{d}t}= {V}_{\beta }- R{i}_{\beta }\end{array}\right.$
where ${i}_{\alpha }$ and ${i}_{\beta }$ are the $\alpha$ and $\beta$ components of the inverter output current vector in the ${\alpha \beta }$ coordinate system. ${V}_{\alpha }$ and ${V}_{\beta }$ are the $\alpha$ and $\beta$ components of the inverter output voltage vector in the ${\alpha \beta }$ coordinate system. The sampling period of the system is set to ${T}_{\mathrm{s}}$. The derivative of the output current in these equations is equated using the forward Euler equations as:
$\left\{\begin{array}{l}\frac{\mathrm{d}{i}_{\alpha }}{\mathrm{d}t}= \frac{{i}_{\alpha }\left({k + 1}\right)- {i}_{\alpha }\left( k\right)}{{T}_{\mathrm{s}}}\\\frac{\mathrm{d}{i}_{\beta }}{\mathrm{d}t}= \frac{{i}_{\beta }\left({k + 1}\right)- {i}_{\beta }\left( k\right)}{{T}_{\mathrm{s}}}\end{array}\right.$
Substitute (4) into (3) and simplify to obtain:
$\left\{\begin{array}{l}{i}_{\alpha }\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{L}{V}_{\alpha }\left( k\right)+ \left({1 -\frac{R{T}_{\mathrm{s}}}{L}}\right){i}_{\alpha }\left( k\right)\\{i}_{\beta }\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{L}{V}_{\beta }\left( k\right)+ \left({1 -\frac{R{T}_{\mathrm{s}}}{L}}\right){i}_{\beta }\left( k\right)\end{array}\right.$
where ${i}_{\alpha }\left( k\right)$ and ${i}_{\beta }\left( k\right)$ are the current components in the ${\alpha \beta }$ coordinate system at ${t}_{k}$ (sampling time $k$); ${i}_{\alpha }\left({k + 1}\right)$ and ${i}_{\beta }\left({k + 1}\right)$ are the predicted values of the current component in the ${\alpha \beta }$ coordinate system at ${t}_{k + 1};{V}_{\alpha }\left( k\right)$ and ${V}_{\beta }\left( k\right)$ are the voltage components in the ${\alpha \beta }$ coordinate system at ${t}_{k}$ sampling times.
When the O-ZSI is in both shoot-through state and non-shoot-through state, we derive the equations for the magnetization current and capacitance voltage flowing through the transformer excitation inductor ${L}_{\mathrm{m}}$ as follows:
1) Shoot-Through States
According to the equivalent circuit in Fig. 2(a), the following circuit equations can be written:
$\left\{\begin{array}{l}{L}_{\mathrm{m}}\frac{\mathrm{d}{i}_{\mathrm{m}}}{\mathrm{d}t}= {v}_{\mathrm{{in}}}- {v}_{\mathrm{C}}\\ C\frac{\mathrm{d}{v}_{\mathrm{C}}}{\mathrm{d}t}= {i}_{\mathrm{m}}\end{array}\right.$
where ${i}_{\mathrm{m}}$ is the magnetization current of the magnetizing inductance ${L}_{\mathrm{m}}$, and the derivative of the magnetization current ${i}_{\mathrm{m}}$ and capacitance voltage ${v}_{\mathrm{C}}$ is approximated by the forward Euler:
$\left\{\begin{array}{l}\frac{\mathrm{d}{i}_{\mathrm{m}}}{\mathrm{d}t}= \frac{{i}_{\mathrm{m}}\left({k + 1}\right)- {i}_{\mathrm{m}}\left( k\right)}{{T}_{\mathrm{s}}}\\\frac{\mathrm{d}{v}_{\mathrm{C}}}{\mathrm{d}t}= \frac{{v}_{\mathrm{C}}\left({k + 1}\right)- {v}_{\mathrm{C}}\left( k\right)}{{T}_{\mathrm{s}}}\end{array}\right.$
Substituting (7) into (6) and simplifying it yield:
$\left\{\begin{array}{l}{i}_{\mathrm{m}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{{L}_{\mathrm{m}}}\left\lbrack {{v}_{\mathrm{{in}}}- {v}_{\mathrm{C}}\left( k\right)}\right\rbrack +{i}_{\mathrm{m}}\left( k\right)\\{v}_{\mathrm{C}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{C}{i}_{\mathrm{m}}\left( k\right)+ {v}_{\mathrm{C}}\left( k\right)\end{array}\right.$
where ${i}_{\mathrm{m}}\left( k\right)$ and ${v}_{\mathrm{C}}\left( k\right)$ are the sampling values of the magnetization current ${i}_{\mathrm{m}}$ and capacitance voltage ${v}_{\mathrm{C}}$ of the transformer magnetizing inductance ${L}_{\mathrm{m}}$ at ${t}_{k}$ and in the shoot-through state, respectively. ${i}_{\mathrm{m}}\left({k + 1}\right)$ and ${v}_{\mathrm{C}}\left({k + 1}\right)$ are the predicted values of magnetization current ${i}_{\mathrm{m}}$ and capacitance voltage ${v}_{\mathrm{C}}$ of the transformer magnetizing inductance ${L}_{\mathrm{m}}$ at $k + 1$ and in the shoot-through state, respectively.
2) Non-Shoot-Through State
According to the equivalent circuit in Fig. 2(b), the following circuit equations can be listed:
$\left\{\begin{array}{l}\left({\gamma - 1}\right){L}_{\mathrm{m}}\frac{\mathrm{d}{i}_{\mathrm{m}}}{\mathrm{d}t}= {v}_{\mathrm{C}}\\\left({\gamma - 1}\right) C\frac{\mathrm{d}{v}_{\mathrm{C}}}{\mathrm{d}t}= \gamma {i}_{\text{inv }}- {i}_{\mathrm{m}}\end{array}\right.$
where ${i}_{\text{inv }}$ is the output current of the O-Z source network. At ${t}_{k},{i}_{\text{inv }}\left( k\right)= {i}_{\mathrm{A}}\left( k\right){S}_{\mathrm{A}}+ {i}_{\mathrm{B}}\left( k\right){S}_{\mathrm{B}}+ {i}_{\mathrm{C}}\left( k\right){S}_{\mathrm{C}}$ can be calculated by sampling the three-phase current at the load side and switching states ${S}_{\mathrm{A}},{S}_{\mathrm{B}}$, and ${S}_{\mathrm{C}}$.
When the ratio of turns is $\gamma = 2$, substitute (7) into (9) and simplify it to obtain:
$\left\{\begin{array}{l}{i}_{\mathrm{m}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{{L}_{\mathrm{m}}}{v}_{\mathrm{C}}\left( k\right)+ {i}_{\mathrm{m}}\left( k\right)\\{v}_{\mathrm{C}}\left({k + 1}\right)= \frac{2{i}_{\text{inv }}\left( k\right)- {i}_{\mathrm{m}}\left( k\right)}{C}{T}_{\mathrm{s}}+ {v}_{\mathrm{C}}\left( k\right)\end{array}\right.$
where ${i}_{\mathrm{m}}\left( k\right)$ and ${v}_{\mathrm{C}}\left( k\right)$ are the sampled values of the magnetization current ${i}_{\mathrm{m}}$ and capacitance voltage ${v}_{\mathrm{C}}$ of the transformer magnetizing inductance ${L}_{\mathrm{m}}$ at ${t}_{k}$ and in the non-shoot-through state, respectively. ${i}_{\mathrm{m}}\left({k + 1}\right)$ and ${v}_{\mathrm{C}}\left({k + 1}\right)$ are the predicted values of the magnetization current ${i}_{\mathrm{m}}$ and capacitor voltage ${v}_{\mathrm{C}}$ of the transformer excitation inductor ${L}_{\mathrm{m}}$ at ${t}_{k + 1}$ and in the non-shoot-through state, respectively.
B. O-ZSI Prediction Function
In the analysis of the O-ZSI discrete model, the transformer is equivalent to the parallel connection of the excitation inductance ${L}_{\mathrm{m}}$ with a set of ideal transformers. ${i}_{\mathrm{m}}$ is the current flowing through the excitation inductance, which cannot be directly measured but can be expressed by ${i}_{\text{inv }}\left( k\right)$ and ${i}_{\mathrm{C}}\left( k\right)$.
In the shoot-through state, the ${N}_{1}$ side is disconnected because diode $\mathrm{D}$ withstands a reverse pressure in the reverse cutoff state, and obtain:
${i}_{\mathrm{m}}= {i}_{\mathrm{C}}= {i}_{\text{inv }}$
Therefore, in the shoot-through state, the predicted values of the transformer excitation inductance current ${i}_{\mathrm{m}}$ and capacitor voltage ${v}_{\mathrm{C}},{i}_{\mathrm{m}}\left({k + 1}\right)$ and ${v}_{\mathrm{C}}\left({k + 1}\right)$, can be rewritten in combination with (8) and (11) as follows:
$\left\{\begin{array}{l}{i}_{\mathrm{m}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{{L}_{\mathrm{m}}}\left\lbrack {{v}_{\text{inv }}- {v}_{\mathrm{C}}\left( k\right)}\right\rbrack +{i}_{\text{inv }}\left( k\right)\\{v}_{\mathrm{C}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{C}{i}_{\text{inv }}\left( k\right)+ {v}_{\mathrm{C}}\left( k\right)\end{array}\right.$
In a non-shoot-through state, it can be concluded that:
$\left\{\begin{array}{l}{i}_{{N}_{1}}= -{i}_{{N}_{2}}/2 \\{i}_{{N}_{2}}= {i}_{\mathrm{C}}- {i}_{\mathrm{m}}\\{i}_{{N}_{1}}+ {i}_{{N}_{2}}+ {i}_{\mathrm{m}}= {i}_{\mathrm{{inv}}}\end{array}\right.$
The above equations can be combined to obtain:
${i}_{\mathrm{m}}= 2{i}_{\text{inv }}- {i}_{\mathrm{C}}$
In the non-shoot-through state, the predicted values ${i}_{\mathrm{m}}\left({k + 1}\right)$ and ${v}_{\mathrm{C}}\left({k + 1}\right)$ can be rewritten by combining (10) and (14):
$\left\{\begin{array}{l}{i}_{\mathrm{m}}\left({k + 1}\right)= \frac{{T}_{\mathrm{s}}}{{L}_{\mathrm{m}}}{v}_{\mathrm{C}}\left( k\right)+ 2{i}_{\text{inv }}\left( k\right)- {i}_{\mathrm{C}}\left( k\right)\\{v}_{\mathrm{C}}\left({k + 1}\right)= \frac{{i}_{\mathrm{C}}\left( k\right)}{C}{T}_{\mathrm{s}}+ {v}_{\mathrm{C}}\left( k\right)\end{array}\right.$
C. O-ZSI Steady-State Model
According to the three-phase symmetrical load parameters in Fig. 1 :
${i}_{\mathrm{{om}}}= \sqrt{\frac{2{P}_{\mathrm{o}}}{3R}}$
where ${i}_{\mathrm{{om}}}$ and ${P}_{\mathrm{o}}$ are the output phase current peak and power, respectively.
The control period is ${T}_{\mathrm{s}}$, the shoot-through time is ${T}_{0}$, and the shoot-through duty cycle is ${D}_{0}= {T}_{0}/{T}_{\mathrm{s}}$. Using the state-space averaging method, the steady-state capacitance voltage ${v}_{\mathrm{C}}$ and magnetizing inductance current ${i}_{\mathrm{m}}$ that flows through the transformer can be expressed as:
$\left\{\begin{array}{l}{V}_{\mathrm{C}}= \frac{{D}_{0}\left({\gamma - 1}\right)}{\gamma {D}_{0}- 1}{v}_{\text{in }}\\{I}_{\mathrm{m}}= \frac{\left({1 -{D}_{0}}\right)\gamma }{1 -\gamma {D}_{0}}{i}_{\text{inv }}\end{array}\right.$
From (17), the DC-link voltage ${U}_{\mathrm{{dc}}}$ of the inverter in the non-shoot-through state can be obtained:
${U}_{\mathrm{{dc}}}= {v}_{\text{in }}- {v}_{{N}_{1}}= {v}_{\text{in }}- \frac{\gamma }{\gamma - 1}{V}_{\mathrm{C}}= \frac{1}{1 -\gamma {D}_{0}}{v}_{\text{in }}= B{v}_{\text{in }}$
where $B$ is the boost factor, and $B = 1/\left({1 -{D}_{0}}\right)$.
According to the conservation of energy, the average input current on the power side can be obtained:
${i}_{\text{in }}= \frac{1 -{D}_{0}}{1 -\gamma {D}_{0}}{i}_{\text{inv }}= \frac{{P}_{\mathrm{o}}}{{v}_{\text{in }}}$
Substituting (19) into (17), obtain the relationship between the magnetized inductor current and the average input current on the power supply side at steady state:
${I}_{\mathrm{m}}= \gamma {i}_{\text{in }}$
III. O-ZSI Improved Sequential-Model Predictive Control
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In this paper, taking the O-ZSI with a transformer as an example, the model predictive control with transformer is solved, and a sequential-model predictive control is designed to eliminate the weighting factors. To solve the model equated as a parallel connection of the magnetizing inductance ${L}_{\mathrm{m}}$ with a set of ideal transformers. The magnetization current ${i}_{\mathrm{m}}$ of the magnetizing inductance is continuous, so the magnetization current ${i}_{\mathrm{m}}$ of the magnetizing inductance can be controlled to control the transformer. The magnetization current ${i}_{\mathrm{m}}$ can be converted predictive control of transformer O-ZSI, the transformer is according to (11) and (15).
A. Traditional FCS-MPC Control Strategy
In the traditional FCS-MPC strategy, the cost function of O-ZSI requires at least three weight factors can be expressed:
$ g ={\lambda }_{{}_{\mathrm{L}}}{\left\lbrack {I}_{{}_{\mathrm{m}\text{ ref }}}\left( k\right)- {i}_{\mathrm{m}}\left( k + 1\right)\right\rbrack }^{2}+ {\lambda }_{{}_{\mathrm{C}}}{\left\lbrack {V}_{{}_{\mathrm{C}\text{ _ref }}}\left( k\right)- {v}_{{}_{\mathrm{C}}}\left( k + 1\right)\right\rbrack }^{2}+ $
${\lambda }_{i}{\left\lbrack {I}_{\alpha \text{_ref }}\left( k\right)- {i}_{\alpha }\left( k + 1\right)\right\rbrack }^{2}+ {\lambda }_{i}{\left\lbrack {I}_{\beta \text{_ref }}\left( k\right)- {i}_{\beta }\left( k + 1\right)\right\rbrack }^{2}$
where ${I}_{\mathrm{m}\text{_ref }}\left( k\right)$ and ${V}_{\mathrm{C}\text{_ref }}\left( k\right)$ is the inductive current and capacitive voltage reference, respectively. ${i}_{\alpha \text{ ref }}\left( k\right)$ and ${i}_{\beta \text{ ref }}\left( k\right)$ are the ${\alpha \beta }$-axis components of the reference output current, ${i}_{\alpha }\left({k + 1}\right)$ and ${i}_{\beta }\left({k + 1}\right)$ are the ${\alpha \beta }$-axis components of the output current prediction; ${\lambda }_{\mathrm{L}},{\lambda }_{\mathrm{C}}$ and ${\lambda }_{\mathrm{i}}$ are the weighting factors for inductor current, capacitor voltage and output current, respectively.
For a three-phase symmetrical load, the corresponding reference value of the control object can be obtained:
$\left\{\begin{array}{l}{i}_{{\mathrm{{inv}}}_{- }\text{ref }}= \sqrt{\frac{2{P}_{\mathrm{o}}}{3R}}\\{V}_{{\mathrm{C}}_{- }\text{ref }}= \frac{{D}_{0}\left({\gamma - 1}\right)}{\gamma {D}_{0}- 1}{v}_{\text{in }}\\{I}_{{\mathrm{m}}_{- }\text{ref }}= \frac{\left({1 -{D}_{0}}\right)\gamma }{1 -\gamma {D}_{0}}{i}_{\text{inv }}\end{array}\right.$
B. The Proposed S-MPC Strategy Without Weighting Factors
It is generally recognized that there is no relevant relationship between the capacitor voltage and the load current, therefore the weighting factor is inevitably required to weigh the two variables. But the tuning of the weighting factors is usually a complex process. The following is the method of eliminating the weighting factors proposed in this paper for the model prediction of O-ZSI.
The cost function in this article is the error between reference value and predicted value:
$\left\{\begin{array}{l}{g}_{\mathrm{m}}= \left|{{I}_{\mathrm{m}\_ \text{ref }}- {i}_{\mathrm{m}}\left({k + 1}\right)}\right|\\{g}_{\mathrm{C}}= \left|{{V}_{\mathrm{C}\_ \text{ref }}- {v}_{\mathrm{C}}\left({k + 1}\right)}\right|\\{g}_{\alpha }= \left|{{i}_{\alpha \_ \text{ref }}- {i}_{\alpha }\left({k + 1}\right)}\right|\\{g}_{\beta }= \left|{{i}_{\beta \_ \text{ref }}- {i}_{\beta }\left({k + 1}\right)}\right|\end{array}\right.$
where ${i}_{\mathrm{m}\text{_ref }}$ is the reference value of the magnetization current of the transformer magnetizing inductance ${L}_{\mathrm{m}}$ at steady state; ${v}_{\mathrm{C}}$ is the reference value of the capacitor voltage at steady state; ${i}_{{\alpha }_{\text{_ref }}}$ and ${i}_{{\beta }_{\text{_ref }}}$ is the $\alpha$ and $\beta$ components of the output current reference values in the ${\alpha \beta }$ coordinate system.
In the traditional sequential-model predictive control (S-MPC1), first, determine n control objects. These objects are then sorted according to their priority, with each control object defining a separate cost function. In the cost function g1, l available switch states are calculated, and the n switch states that minimize the cost function g1 are selected for regulating the first control object.
In the cost function g2, all n available switch states are calculated, and the $n - 1$ switch states that minimize the cost function g2 are selected for regulating the second control object.
In the cost function ${g}_{\mathrm{x}- 1}$, all $n - x + 3$ available switch states are calculated, and the $n - x + 2$ switch states that minimize the cost function ${g}_{\mathrm{x}- 1}$ are selected for regulating the $x - 1$ control object.
In this paper, capacitor voltage and the ${\alpha \beta }$-axis current component of the output current, which are selected as control objects, switching states equal to ${2}^{3}$ available switching states are calculated, and n is the control objects equals to three, removing one of the shoot-through state variables, so l equals to seven. Therefore, seven available switch states are calculated, and the three switch states that minimize the cost function ${g}_{\mathrm{c}}$ are selected for the next cycle. In the cost function ${g}_{\alpha }$, three available switch states are calculated, and the two switch states that minimize ${g}_{\alpha }$ are selected. This process is repeated in ${g}_{\beta }$, where two available switch states are calculated, and the switch state that minimizes ${g}_{\beta }$ is selected.
The proposed sequential-model predictive control (S-MPC2) can be realized by the following steps, first identify n control objects and $m$ switching states, each control object defines a separate cost function that is computed sequentially.
In this paper, n is the control objects also equals to three, removing one of the shoot-through state variables, so $m$ equals to seven. In order not to increase the computational burden and to have enough voltage vectors to be analyzed, a fixed-step formula was proposed and the optimal result is equals to two that obtained by screening the step size:
$ S =\left({m - 1}\right)/n $
Therefore, seven available switch states are calculated, and the five switch states that minimize the cost function ${g}_{\mathrm{c}}$ are selected for the next cycle. In the cost function ${g}_{\alpha }$, five available switch states are calculated, and the three switch states that minimize ${g}_{\alpha }$ are selected. This process is repeated in ${g}_{\beta }$, where three available switch states are calculated, and the switch state that minimizes ${g}_{\beta }$ is selected.
Based on the mathematical model of O-ZSI and advantages of S-MPC, this paper designs the O-ZSI sequential-model predictive control. The algorithm execution flowchart is shown in Fig. 3. The magnetization current ${i}_{\mathrm{m}}$ of the excitation inductor decreases when the O-ZSI is in the non-shoot-through state and increases when it is in the shoot-through state. Based on this characteristic, this paper determines whether the next control cycle is in the shoot-through state according to the predicted magnetization current ${i}_{\mathrm{m}}$. First, using the cost function ${g}_{\mathrm{m}}$ in (23), we determine whether the next switching state of the inverter is in the shoot-through state based on the predicted ${i}_{\mathrm{m}}\left({k + 1}\right)$ of the magnetization current ${i}_{\mathrm{m}}$ of the transformer magnetizing inductance ${L}_{\mathrm{m}}$ in the shoot-through state and non-shoot-through state.
In the shoot-through state, all switches of the inverter bridge are turned on. In seven non-shoot-through states, first, the predictive capacitor voltage is first calculated using (15); then, the cost function ${g}_{\mathrm{C}}$ of the capacitor voltage in (23) is calculated. The two switching states that make the capacitor voltage most costly are eliminated from the seven non-shoot-through states. The remaining five non-shoot-through switching states are temporarily selected. Using (5), the predicted output current in the remaining five non-shoot-through states is calculated. Then, the cost function ${g}_{\alpha }$ of the load current in (23) is calculated, eliminating the two switching states that make the load current $\alpha$-axis component the costliest. Among the remaining three switching states, the $\beta$-axis component cost function is calculated using (23), and the two switching states that maximize the cost are eliminated, and the remaining optimal non-shoot-through switching state is selected. Finally, the remaining optimal non-shoot-through switching state is selected and applied to the next control cycle of O-ZSI.
The above calculations show that the seven switching states have different effects on control effects on the control quantity, and the two switching states with the largest error compared to the expected value are removed by sorting them from the most effective to the least effective, and since n equals to three, the above steps are carried out three times, and the cost functions for the capacitor voltage cost function and the load current ${\alpha \beta }$ are computed in turn, retaining the rest of the switching states that have a small difference from the expected value.
Since the proposed strategy removes the switching states that make the current cost function the largest, the particularly unsuitable switches have been removed, and the switching states are used as one of the two switching states that can be optionally removed in the conventional cost function, the proposed strategy can effectively reduce the computation burden. The control block diagram of S-MPC algorithm for O-ZSI is shown in Fig. 4.
Fig. 5 is depicted as an example of the voltage vectors used in each step. The first step 1 is removing the two states with the maximum ${g}_{\mathrm{c}}$ values. The next step 2 is removing the two states with the maximum ${\mathrm{g}}_{\alpha }$ values. The last step 3 is removing the two states with the maximum ${g}_{\beta }$ values.
IV. Simulation Analysis
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To verify the feasibility and effectiveness of the proposed sequential-MPC strategy (S-MPC2) for O-ZSI in this paper, the simulation analysis is performed using MATLAB/Simulink and compared with the traditional FCS-MPC and traditional S-MPC strategy (S-MPC1). The simulation parameters are shown in Table II, and the weighting factors of the traditional FCS-MPC in the table are obtained by extensive debugging.
A. Steady-State Performance
Considering the O-ZSI stability under different operating conditions, the output current total harmonic distortion (THD), root mean square error (RMSE) of capacitor voltage, and RMSE of output current in ${dq}$-axis are tested respectively to evaluate the control performance. By measuring and evaluating these indicators, we can assess the steady-state performance of the different model predictive control strategy for O-ZSI. Fig. 6 compares the control performance of three strategies, FCS-MPC, S-MPC1, and S-MPC2, under different output power conditions.
The performance of S-MPC2 and FCS-MPC in terms of output current THD is found to be comparable. Additionally, the overall trend of THD increases with an increase in the reference power, while exhibiting minimal fluctuation. On the other hand, S-MPC1 shows consistently higher THD values and greater volatility compared to FCS-MPC and S-MPC2.
The RMSE of capacitor voltage shows that S-MPC1 exhibits a relatively larger error, whereas FCS-MPC and S-MPC2 demonstrate similar trends. However, under most conditions, S-MPC2 exhibits slightly better performance than FCS-MPC. Analyzing the RMSE of output current reveals that FCS-MPC is relatively stable compared to S-MPC2, with a smaller fluctuation range for the ${I}_{d\text{-axis }}$ component. Conversely, S-MPC2 shows less fluctuation for the ${I}_{q\text{-axis }}$ output current component.
In a word, the simulation results indicate that S-MPC2 strategy can maintain comparable results to FCS-MPC while reducing computational complexity. In contrast, S-MPC1 shows inferior performance despite its lower computational complexity. Both S-MPC strategies exhibit certain advantages in terms of computational complexity, with S-MPC2 outperforming S-MPC1 in overall performance.
B. Power Step-Change
Letting the input voltage of the system ${v}_{\text{in }}= {100}\mathrm{\;V}$ and the DC-link voltage ${U}_{\mathrm{{dc}}}= {200}\mathrm{\;V}$.(17) can calculate ${V}_{\mathrm{C}\text{_ref }}= -{50}\mathrm{\;V}$, and (18) can calculate the shoot-through duty cycle ${D}_{0}= {0.25}$. When the output power reference value ${P}_{\mathrm{o}}$ ref abruptly changes from 1000W to 540W, the load phase current peak reference value ${i}_{\text{om }- \text{ref }}$ changes from 8.165A to 6A.
Fig. 7(a) is the simulation waveform under the traditional FCS-MPC strategy, and Fig. 7(b) is the simulation waveform of the S-MPC2 proposed in this paper. When the output power step occurs in 0.20 s, the output phase current rapidly follows the reference value under the two strategies, and the peak value changes from 8.165A to 6A, while the capacitor voltage basically remains at -50V. Both strategies achieve model predictive control of O-ZSI and can resist the output power disturbance ability, which can ensure that the system can reliably and stably work when there is this disturbance.
The FFT analysis of the output A-phase current before 0.20s in Fig. 8 shows that the harmonic distortion rate is 2.96% and 2.68% with the traditional FCS-MPC strategy and S-MPC2 strategy, respectively. The output current waveform quality is better under the proposed sequential-MPC strategy.
C. Frequency Step-Change
Let the input voltage of the system be ${v}_{\text{in }}= {100}\mathrm{\;V}$, the DC-link voltage be ${U}_{\mathrm{{dc}}}= {200}\mathrm{\;V}$ unchanged, and the output power reference value be ${P}_{{o}_{- }\text{ref }}= {540}\mathrm{\;W}$ when the output frequency reference value changes from 50Hz to 100Hz
Fig. 9(a) is the simulation waveform in the traditional FCS-MPC strategy, and Fig. 9(b) is the simulation waveform of the S-MPC2 proposed in this paper. When the output frequency reference value step occurs at 0.20s, the capacitor voltage remains stable with the two strategies, and the load three-phase current can follow the reference value well, which indicates that both FCS-MPC strategies are resistant to output frequency disturbance.
V. Experimental Verification
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To verify the feasibility of the proposed S-MPC strategy for O-ZSI, an experimental prototype is built, as shown in Fig. 10. The parameters of each component in the main circuit are as follows: the capacitance is $C ={1000\mu }\mathrm{F}$ in the O-Z source topology, the transformer uses a customized high-frequency transformer with low leakage inductance and a 2:1 turn ratio, the IGBT power switch is an FF100R12RT4 from INFINEON, and the filter inductance ${L}_{\mathrm{a}}= {L}_{\mathrm{b}}= {L}_{\mathrm{c}}= {10}\mathrm{{mH}}$. The control algorithm is implemented based on the dSPACE platform. To avoid electromagnetic interference, the dSPACE output control signal is transmitted to the drive circuit via an optical fiber, which controls the IGBTs.
A. Steady-State Performance
Given the input voltage ${v}_{\text{in }}= {100}\mathrm{\;V}$ and DC-link voltage ${U}_{\mathrm{{dc}}}= {200}\mathrm{\;V}$, and the output power reference value ${P}_{\mathrm{o}\text{ ref }}= {540}$ W, the experimental results are shown in Fig.11. In Fig.11(a) and Fig. 11(c), ch1 shows the experimental waveform of the capacitor voltage, ch2 shows the experimental waveform of the DC-link voltage, ch3 shows the experimental waveform of the unfiltered ac voltage, and ch4 shows the experimental waveform of the output current. All of them are well controlled, the proposed sequential-MPC strategy does not need to adjust the weight factors, and the FFT analysis in Fig. 11(b) and Fig. 11(d) show that the THD of the output current of the traditional FCS-MPC strategy is smaller than that of sequential-MPC strategy when on steady-state. However, the difference is not big, yet the calculation is simpler, proving that the model prediction using the weight elimination technique can also basically achieve the effect of the traditional FCS-MPC.
B. Power Step-Change
Given the input voltage ${v}_{\text{in }}= {100}\mathrm{\;V}$ and $\mathrm{{DC}}$-link voltage ${U}_{\mathrm{{dc}}}=$ ${200}\mathrm{\;V}$, when the output power reference ${P}_{\text{o_ref }}$ changes from 1000W to 540W, the experimental results are shown in Fig. 12. In both strategies, the capacitor voltage remains unchanged and stable at approximately 50V, the DC-link voltage remains unchanged and is boosted to approximately 200V, and the output current abruptly changes from 8.165A to 6A. It is evident that the two strategies in the output power steps state, The control quantities can track the reference value well, and the waveforms match those in the simulation analysis after the step-change in power.
C. DC-Link Voltage Step-Change
Given the input voltage ${v}_{\mathrm{{in}}}= {100}\mathrm{\;V}$ and the output power reference value ${P}_{{\mathrm{o}}_{- }\text{ref }}= {540}\mathrm{\;W}$. When the DC-link voltage reference value abruptly changes from 100V to 80V, the experimental results are shown in Fig. 13. The capacitor voltage and the DC-link voltage can be kept stable in the traditional FCS-MPC strategy and the proposed sequential-MPC strategy. It can be seen that under this condition, the proposed sequential-MPC can also achieve similar results as the FCS-MPC strategy.
D. Frequency Step-Change
In this experiment given the input voltage ${v}_{\text{in }}= {100}\mathrm{\;V}$, the DC-link voltage ${U}_{\mathrm{{dc}}}= {200}\mathrm{\;V}$ remains unchanged, and the output power reference value ${P}_{{o}_{\_ }\text{ref }}= {540}\mathrm{\;W}$, when the output frequency reference value abruptly changes from 50Hz to 100Hz. The experimental results are shown in Fig. 14. In both strategies, the capacitor voltage and the DC-link voltage can follow the reference value well. Each control quantity can also track the reference value well in the frequency step-change case, which is consistent with the waveform of the simulation analysis after the sudden change in frequency. The FFT analysis shows that the quality of the output current of the proposed sequential-MPC strategy is better than that of the traditional FCS-MPC strategy, both before and after the sudden change in frequency.
VI. Conclusion
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To solve the problem of FCS-MPC for O-ZSI with the transformer, this paper proposed an improved O-ZSI sequential-MPC strategy without weighting factors. The feasibility and effectiveness of the proposed strategy are verified by simulation analysis and experiment results.
1) The strategy proposed in this paper effectively solves the model predictive control with the transformer Z-source topology and can achieve closed-loop control, simple adjustment, and easy digital implementation.
2) Compared with the traditional FCS-MPC and traditional S-MPC strategy, the proposed sequential-MPC strategy in this paper can balance the weights of each control variable without adjusting the weighting factors. And it has comparable when on steady state and better dynamic properties, but makes the calculations easier.
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Appendix
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Year 2024 volume 9 Issue 2
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doi: 10.24295/CPSSTPEA.2024.00002
  • Receive Date：2023-05-30
  • Online Date：2025-07-05
  • Published：2024-06-10
Article Data
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  • Received：2023-05-30
  • Revised：2024-01-02
  • Accepted：2024-01-29
Affiliations
    University of Shanghai for Science and Technology Department of Electrical Engineering Shanghai China

Corresponding:

Wei Luo.
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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