RNA Identification Technique and RST Control of a Hybrid Indirect Matrix Converter with a Flying Capacitor Three Level Inverter in Power Active Filtering Application

The quality of electrical energy concerns all actors in the energy sector, namely producers, distributors and consumers of electricity. It has become an important subject in recent years, because of electrical disturbances have a high cost for manufacturers because they cause production stoppages, losses of raw materials, a drop in the quality of production, premature aging of equipment. Power electronics, which are sensitive to disturbances or disturbance generators such as power switches [1].

Among the power electronic devices present on the electrical network, some converters are used in active filtering applications. Their action consists of inject currents into the network in order to improve the quality of voltages and currents available. Improving the performance of these devices under various conditions operation of energy sources and loads in the electrical network is a constant concern of certain players in the research world [2].

Harmonic current caused by non-linear loads reduce the PG quality. One of the several introduced solutions to compensate this harmonic is the active power filters (APF), The APF is composed of a transformer connects the grid to an AC-DC converter through a third order RLC filter [3], this converter feed a two capacitor filter to provide two separated voltage sources for the Flying Capacitor Inverter (FCI) and an RL filter.

This study is focused on the synthesize of an RST controller for the FCI and its output RL filter, we interest in the reference current tracking even when there is a perturbation on the FCI input DC voltage, for that reason, a transformer followed by a third order RLC filter as mentioned in Ref. [4] and a three phase full wave converter is used to provide a perturbed input DC voltage to the inverter. The FCI must generate an equal currant to the harmonics in an opposite phase, and it semi-conductors switches must support the voltage applied on it, switches dynamic represent the inverter time response, this dynamic has a reverse relation with the supported voltage [2].

To enhance the filtering quality an output passive filter recorded the inverter to PG is dimensioned to stop propagation of harmonics produced by the FCI to PG.

The FCI who belong to the serial multi-cellular inverter family had a topology that reduced the applied voltage on the switches, which let us pick a fast dynamic switch and allowed to used it in both middle and high PG voltage, moreover, these inverters harmonics are from rang f_c×N (f_c: carrier frequency; N: cell number) when we used carriers evenly phase shifted [5], and that reduced the constraints on the output filter. In this case a simple RL filter can satisfy the passive filter objective.

The commonly harmonics identification technique used is that named phase-locked loop (PLL) [4], as it exists several other methods such as wavelet [6], Kalman filtering (KF) [7], instantaneous power \cite{b5}, synchronous reference frame [8], fast Fourier transformation [9, 10] and ANN [11, 12].

An ANN of Adaline type which has an excellence performance in signal estimation [12] is introduced to extract the harmonic currant, this ANN type is proposed and developed by Widrow and Walach [13], and targets are the Fourier components amplitude of the harmonic currant.

This work is organized as follows: section Ӏ discusses in brief the active filtering process structure and parts; section IӀ for the RST controller synthesize method which done in the open and closed loop; section III results, discussion and comparison between both synthesize methods; section IV conclusion.

The Flying Capacitor Multilevel Inverter (FCMI) is considered as a regulated voltage source that is controlled in an open loop with the saw-tooth rotation PWM (STRPWM), it is considered as a first order system with less than one cycle of PWM duration delay, FCMI and its output RL filter represent a second order system which is controlled in a closed loop with an RST controller, the reference of this last loop is delivered by Adaline identification technique, full loop concept is schematized in Figure 1.

1.png

Figure 1. Control structure

An RST linear controller is implemented in order to eliminate the error between I_ref(t) and I_inj(t). This polynomial controller provides more freedom degree then the classical PI controller [4]. The transfer function (TF) of the loop shown in Figure 5.

5.png

Figure 5. Control loop with RST controller

$\begin{align}  & {{I}_{inj}}(s)=\frac{TB(S)}{(SA+RB)(s)}{{I}_{REF}}(s) \\ & +[\frac{S(S)}{(SA+RB)(s)}(\frac{1}{l}-(\frac{r}{l}+s))]{{V}_{s}} \\\end{align}$          (11)

When A(s) and B(s) are the full system denominator and nominator respectively,

$\begin{align}  & A(s)=(s+\tau )(s+r/l) \\ & B(s)=1/(l\tau ) \\\end{align}$       (12)

ΔV represent the inverter injected harmonic, and it frequency is fc*N which is filtered by the RL output filter. For that reason, we ignored it in the TF.

$D(s)=(SA+RB)(s)$              (13)

Eq. (13) is called Bezout equation. For the purpose of achieving regulation and robustness objectives we have:

The RST controller causality condition:

$\begin{align}  & Deg(S)\ge Deg(R) \\ & Deg(S)\ge Deg(T) \\\end{align}$           (14)

As we have:

$Deg(A)\ge Deg(B)$            (15)

We got:

$Deg(D)=Deg(S)+Deg(A)$                 (16)

Perturbation (V(s)) rejection condition:

$Deg(S)=0$                 (17)

It is a simplified condition, for preciseness calculation of S(f)=0 we take f=50, f is the perturbation frequency.

Tracking condition:

$T(0)=R(0)$               (18)

The unknown parameters number (UPN) defined from Bezout equation:

$UPN=\deg (S)+\deg (R)+1$                 (19)

And for controlling all Bezout equation parameters:

$deg(R)\ge deg(A)$                    (20)

To get a unique solution:

$deg(R)=deg(A)=n$                (21)

For a (proper) controller:

$deg(S)=deg(R)$                 (22)

And it yields:

$deg(D)=2\times n$                  (23)

For a strictly proper controller we modify Eq. (22):

$deg(S)=deg(R)+1$              (24)

And it yields:

$deg(D)=(2\times n)+1$             (25)

In our application we chose a proper controller, and our strategy to define the unknown parameters is:

Elimination of system dynamic:

$deg(R)=deg(A)\Rightarrow deg(R)=2$                     (26)

And with no influence on the reference

$T(s)=R(s)$                 (27)

So the final loop with a rejected perturbation (V(s)) is illustrated in Figure 6:

6.png

Figure 6. The resulted equivalent control loop

Then we impose our desired dynamic:

$\deg (S)=2\Rightarrow S(s)={{s}_{2}}{{S}^{2}}+{{s}_{1}}S$                   (28)

S(s) parameters calculated either in an open loop (OL) as in a closed loop (CL).

First, CL controller synthesize: CL transfer function is:

${{F}_{cl}}(s)=\frac{1/(l\times \tau )}{{{s}_{2}}{{S}^{2}}+{{s}_{1}}S+1/(l\times \tau )}$                (29)

The desired denominator D_desi(s) is:

${{D}_{desi}}(s)={{S}^{2}}+2\xi {{\omega }_{0}}S+\omega _{0}^{2}$                (30)      

where, ξ= 0:7 which represent the best tradeoff between the time response t_r and the pulsation ω₀;

ω₀ is defined to satisfied the condition:

$-{{5}_{DB}}\ge {{G}_{DB}}\ge -{{10}_{DB}}$         (31)  

[-10_DB, -5_DB] is the interval that gives the acceptable ratio between performance and frequency filtering (robustness),

${{G}_{DB}}={{\left| {{F}_{cl}}(j{{\omega }_{f}}) \right|}_{DB}}$            (32)

ω_f  is the pulsation that we’re up to filtrate and it represent the harmonics produced by the inverter from range fc×N. It is complicated to calculate the last equation, so we used the asymptote P(s) of F_cl in high frequency,

$\left| P(j\omega ) \right|={{(\frac{{{\omega }_{0}}}{\omega })}^{2}}$           (33)

then we replaced ­|F_cl(ω_f)| by |P(ω_f)| in Eq. (32), and Eq. (31) we got:

${{s}_{2}}={{(\frac{1}{\omega _{0}^{2}l\tau })}^{2}}$           (34)

${{s}_{1}}=2\xi {{\omega }_{0}}{{s}_{2}}$             (35)

Second, OL controller synthesize: OL transfer function is:

${{F}_{ol}}(s)=\frac{1/(l\times \tau )}{{{s}_{2}}{{S}^{2}}+{{s}_{1}}S}$           (36)

We know that both open and closed loop had the same response in high frequency, for this reason we used the condition Eq. (31) same as in CL, but instead of Eq. (32) we have:

${{G}_{DB}}={{\left| {{F}_{ol}}(j{{\omega }_{f}}) \right|}_{DB}}$             (37)

Eq. (37) developed to:

${{s}_{2}}^{2}\omega _{f}^{4}+{{s}_{1}}^{2}\omega _{f}^{2}-{{(\frac{B}{{{G}_{DB}}})}^{2}}=0$            (38)  

To reach tracking in finite time we have:

$\Delta {{\varphi }_{desired}}=180+{{\left| {{F}_{ol}}(j{{\omega }_{c}}) \right|}_{DB}}$            (39)

This gives:

$\frac{{{\omega }_{c}}{{S}_{2}}}{{{S}_{1}}}=\tan (90-\Delta {{\varphi }_{desired}})$             (40)

ω_c is the pulsation of |F_cl(ω_f)|_DB intersection with 0_DB line,

$\left| {{F}_{ol}}(j{{\omega }_{c}}) \right|=1$     (41)

And it developed to:

${{s}_{2}}^{2}\omega _{c}^{4}+{{s}_{1}}^{2}\omega _{c}^{2}-{{B}^{2}}=0$            (42)

The desired phase margin condition for an accepted time response:

$60{}^\circ \ge \Delta {{\varphi }_{desired}}\ge 45{}^\circ $           (43)

Eq. (42) admit one acceptable solution value of ω_c,

$\omega _{c}^{2}=\frac{-{{s}_{1}}^{2}+\sqrt{{{s}_{1}}^{4}+4{{({{s}_{2}}B)}^{2}}}}{2{{s}_{2}}^{2}}$          (44)

the other solution is negative, which is not an acceptable value. ω_cs₂ is replaced by its value from Eq. (40) in Eq. (44),

$\begin{align}  & {{S}_{2}}=\frac{{{S}_{1}}^{2}\eta }{B} \\ & \eta =\tan (90-\Delta {{\varphi }_{desired}})\sqrt{1+{{(\tan (90-\Delta {{\varphi }_{desired}}))}^{2}}} \\\end{align}$       (45)

Then we replace Eq. (45) in Eq. (38), it gives one acceptable solution for S₁²:

${{S}_{1}}^{2}=\frac{\frac{{{B}^{2}}}{{{\eta }^{2}}}+\sqrt{\frac{{{B}^{4}}}{{{\eta }^{4}}}+\frac{4{{B}^{4}}}{{{(\eta \times {{G}_{DB}})}^{2}}}}}{2\omega _{f}^{2}}$      (46)   

It is clear that we start our calculations with s₁ from Eq. (46), and then we passed to calculate s₂ from Eq. (45).

In this paper a precise RST controller synthesize methods is presented, first method based on closed loop transfer function in addition of an asymptote, while the second method based on the open loop transfer function, both methods designed to satisfied fast harmonic tracking, robustness and rejects the FCMI switching harmonics. Also, the integration of the Adaline ANN shows an accurate and fast response to provide reference current to the controller. The maintain of the flaying capacitors voltages attains by using the STRPWM and the natural stability of the active power filter. Results obtained by simulation in MATLAB/Simscap/Sim Power System shows efficiency of these methods to ensure a reference finite time tracking in the filtration application.