Control for the currents balancing of a multicell
interleaved converter with ICT

Cyrille Gautier, Fabien Adam, Eric Labouré, Bertrand Revol, Denis Labrousse

► To cite this version:

Cyrille Gautier, Fabien Adam, Eric Labouré, Bertrand Revol, Denis Labrousse. Control for the currents balancing of a multicell interleaved converter with ICT. 2013 15th European Conference on Power Electronics and Applications (EPE), Sep 2013, Lille, France. 10.1109/EPE.2013.6631991. hal-01686620

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# Control for the currents balancing of a Multicell Interleaved
## Converter with ICT

Cyrille Gautier¹, Fabien Adam¹, Eric laboure², Bertrand Revol¹, and Denis Labrousse¹

SATIE, ENS Cachan, CNRS, UniverSud (¹)

61 av. du President Wilson

94 230 Cachan, France

LGEP, CNRS, UniverSud (²)

11, rue Joliot Curie, Plateau de Moulon

91192 Gif sur Yvette, France

firstname.lastname@satie.ens-cachan.fr (¹)

firstname.lastname@lgep.supelec.fr(²)

## Keywords

<<Interleaved converters>>, <<Multilevel converters>>, <<Converter control>>, <<DSP>>.

## Abstract

Toward the aim of advanced integration in power electronic, the properties of Multicell interleaved converters are of prime interest. Among these properties, the EMC behavior is interesting and the dynamic performances are excellent. However their implementation requires the monitoring and balancing of intercell currents. On one hand, the converter control must ensure a satisfying balanced distribution of currents among switching cells outputs, so that magnetic components are used optimally. On the other hand, this control must also ensure the regulation loop of the output current. To achieve these goals, we decided to work on a direct matrix model, which allows to highlight the fast eigenmode related to the power (common mode) and the slow magnetizing eigenmodes (differential modes). The synthesis of the control and the calculus of corrector parameters are performed using inversion of this direct model. A digital control based on this method has been implemented, and the measurement results that we present confirm it's efficiency on current balancing.

## Introduction

Among the multilevel converter typologies that have been presented in the literature [1], those based on multiphase half bridge legs with coupled inductors at the output stage are of particular interest toward dynamic and integration abilities [2]. The DC/AC converter topology on which we are working is made of several elementary half bridge inverters connected in parallel, switching at the same frequency f. The control of these elementary cells is interleaved and their outputs are magnetically coupled. This magnetic coupling between the outputs of the switching cells allows to reduce current constraints on power devices, compared to the case of a multiphase converter where the outputs are connected through non-coupled self-inductances. As presented in [3], the magnetic coupling can be achieved through the use of one monolithic magnetic core, or by the association of separate, identical two-windings intercell transformers (ICT) [4]. Moreover, EMC behavior of multicell interleaved converter using ICT is very interesting compared to non-interleaved structures [6]. Previous works have shown that the solution of using separate identical elementary transformers ensures a high level of integration of the magnetic part [5]. It also allows to keep a versatile topology regarding the number q of parallel switching cells, thus several converters with different number of cells can be studied.

The control of current balancing has to ensure that there is no drift of the magnetic flux in the transformers cores, and the aim of the work presented here is to allow accurate control of this structure in order to optimize control algorithms. For this purpose, we set up a direct linear model of the converter by performing a modal decomposition of the currents in the magnetic coupling part. At last, a chapter will be dedicated to the experimental bench and to the implementation of the digital controller in DSP. There, we will discuss on experimental results.
<figure><img src="image_2.png" /><figcaption>Figure 1: Multicell Inverter with ICT</figcaption></figure>

current balancing

In the case of q switching cells, the ICT is build around q identical two-winding transformers connected in a cyclic-cascade topology, as presented figure 2. It is mandatory to ensure the $i_q$ currents balance between the different paths in structures where power devices are connected in parallel. This constraint is especially important in the case of power structures with ICT because drift current of one channel can cause saturation of one magnetic core of the ICT which, as a consequence, leads to over-currents. Although the different power pathways are similar, the drift of the currents can occur due to the natural dispersion of electric and magnetic characteristics of these pathways. The use of separate ICTs inevitably leads to such dispersion, and a special attention should be paid to the ICT placement and to the balance between connection lengths.

<figure><img src="image_3.png" /><figcaption>Figure 2: ICT: cyclic-cascade topology</figcaption></figure>

For instance, considering the equivalent resistance *R* as measured at the output of one phase of the converter. The equivalent circuit of the resistances at the outputs of the phases is presented figure 3. Duty cycle is the same in all phases, equal to *α*, which can be written *α*<sub>k</sub> = *α*, k ∈ [1, q]. A small variation *ΔR* in one phase resistance compared to the others (equation 1) will cause a current drift in this phase as expressed in equation 2.

<figure><img src="image_4.png" /><figcaption>Figure 3: Current unbalance versus variation of one resistance</figcaption></figure>
$$ \left\{ \begin{array}{l} R_1 = R + \Delta R \\ R_k = R, k \in [2, q] \end{array} \right. \qquad (1) $$

$$ \frac{\Delta I_1}{I_1} = - \frac{q-1}{q} \frac{\Delta R}{R} \qquad (2) $$

A drift of current can also occur because of jitter on duty cycles $\alpha_k$, linked to digital time discretization of drivers signals. The active control toward current balancing that we present in this work is intended to minimize these effects, and the control model of the converter presented in the following part has been built in this goal.

At first, we have defined a direct model of the converter in an orthogonal basis where only normal modes exists. As a consequence, the ICT impedance matrix defined in this basis is diagonal, and each diagonal element relies on the time constant of one normal mode. Secondly, this model has been inverted and the balancing control has been defined.

## Converter model

### Direct linear model

We use an averaged model over the switching period, set to $T_s = 100 \mu s$ in our experimental bench. The relationship between voltages and currents regarding the magnetic coupling device is described through the $[Z(s)]$ impedance matrix in the s-Domain, as defined using equation 3. Measurement of the electrical characteristics of each ICT gives practically equivalent results. Assuming that they remain in a linear mode of operation, the impedance matrix $[Z(s)]$ of the ICTs is then given by the equation 4.

$$ \begin{pmatrix} V_1 - V_s \\ \vdots \\ V_k - V_s \\ \vdots \\ V_q - V_s \end{pmatrix} = [Z(s)] \cdot \begin{pmatrix} I_1 \\ \vdots \\ I_k \\ \vdots \\ I_q \end{pmatrix} \qquad (3) $$

$$ [Z(s)] = \begin{bmatrix} R + 2L_s & -M_s & 0 & \cdots & 0 & -Ms \\ -M_s & R + 2L_s & -M_s & 0 & \cdots & 0 \\ 0 & \cdots & \cdots & \cdots & \cdots & \vdots \\ \vdots & \cdots & \cdots & \cdots & \cdots & 0 \\ 0 & \cdots & 0 & -M_s & R + 2L_s & -M_s \\ -M_s & 0 & \cdots & 0 & -M_s & R + 2L_s \end{bmatrix} \qquad (4) $$

The R term is the sum of a DC resistor component and of a losses component, for one phase; L and M are the self-inductance and the mutual-inductance of one two-winding transformer of the ICT, which can be seen on figure 4. The non-diagonal terms of the $[Z(s)]$ matrix represent the magnetic coupling effects, and the main idea of the proposed model is to find a new basis where the relationship between voltages and currents would be described through a diagonal matrix, highlighting the normal modes of the converter.

### Change of basis

We want that the synthesis method remains general and independent of the number of cells. The diagonal matrix $[H_d(s)]$ is calculated through eigendecomposition of the matrix $[Z(s)]$, as defined in equation 5, where $[P]$ is the square $q \times q$ matrix of eigenvectors. Equation 6 allows the calculation of $I^*$ and $V^*$, which are the current and voltage vectors in the new orthogonal basis.

$$ [H_d(s)] = P^{-1} \cdot [Z(s)] \cdot P \qquad (5) $$
$$ \left\{ I^* = P^{-1} \cdot \begin{pmatrix} I_1 \\ \vdots \\ I_k \\ \vdots \\ I_q \end{pmatrix} ; \quad V^* = P^{-1} \cdot \begin{pmatrix} V_1 - V_s \\ \vdots \\ V_k - V_s \\ \vdots \\ V_q - V_s \end{pmatrix} \right. \qquad (6) $$

### Case of q=6 cells

We will illustrate this principle by applying it to the particular case of an interleaved converter of q=6 cells, which is the configuration of our prototype. The transformation matrix $[P^{-1}]$ is then given by equation 7.

$$ P^{-1} = \frac{1}{\sqrt{6}} \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ -1 & 2 & -1 & -1 & 2 & -1 \\ -1 & -1 & 2 & -1 & -1 & 2 \\ 1 & -1 & 2 & -1 & 1 & 2 \\ -1 & -2 & -1 & 1 & 2 & 1 \\ -1 & 1 & -1 & 1 & -1 & 1 \end{bmatrix} \qquad (7) $$

Equation 8 describes the $[H_d(s)]$ matrix we obtained in this case. $L, M$ and $R$ values have been measured on the two-winding transformers ($L = 27.0 \, mH, M = 26.7 \, mH$ and $R = 0.45 \, \Omega$) and they lead to the time constants $\tau_1, \dots, \tau_4$. These time constants are given in table I.

$$ H_d(s) = \frac{1}{R} \begin{bmatrix} \frac{1}{1+\tau_1 s} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{1+\tau_2 s} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{1+\tau_2 s} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{1+\tau_3 s} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{1+\tau_3 s} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{1+\tau_4 s} \end{bmatrix} \qquad (8) $$

<table><caption>Table I: Time constants of normal modes</caption><thead><tr><th></th><th>expression</th><th>value (ms)</th></tr></thead><tbody><tr><td>τ<sub>1</sub></td><td><sup>2(L - M)</sup>⁄<sub>R</sub></td><td>0.667</td></tr><tr><td>τ<sub>2</sub></td><td><sup>2L + M</sup>⁄<sub>R</sub></td><td>87.7</td></tr><tr><td>τ<sub>3</sub></td><td><sup>2L - M</sup>⁄<sub>R</sub></td><td>30.0</td></tr><tr><td>τ<sub>4</sub></td><td><sup>2(L + M)</sup>⁄<sub>R</sub></td><td>119</td></tr></tbody></table>

Large differences between these time constants must be noticed. Especially, the first and smallest time constant $\tau_1$ is linked to the fastest normal mode of the converter. As the first row of the matrix $[P^{-1}]$ is filled with 1, equation 9 shows that this mode is related to the converter output current $I_s$, defined figure 1, which is also the common-mode current in this case.
$$I_1^* = \frac{1}{\sqrt{6}} \cdot (I_1 + I_2 + I_3 + I_4 + I_5 + I_6) = \frac{I_5}{\sqrt{6}} \qquad (9)$$

However, $\tau_1$ remains large enough compared to the switching period to allow the use of an averaged model. This first result is interesting as it highlights and confirms the fast dynamic response behavior of this kind of converters.

The others time constants $\tau_2, \dots, \tau_4$ correspond to much more slower modes. Since the sum of the elements of the corresponding row of the $[P^{-1}]$ matrix is zero, these time constants are related to differential mode of magnetic flux inside ICTs. The paths of the common mode (CM) and differential mode (DM) of the magnetic flux in one two-winding transformer are shown on figure 4, and the magnetization of the ICT core is partly made by these magnetic flux of differential mode. The path of the common mode magnetic flux returns via the air, which is a high reluctance path. Then, the magnetic leakage are required for the good operation of the structure, and they have to be accurately determined during the design of the ICTs. If we look at the path of the magnetic flux of differential mode in one transformer, it only follows the magnetic core, as shown on the right part of the figure 4. Indeed, this is a path of low reluctance, so a small lack of balancing between currents may cause high induction levels. The normal differential modes of the structure defined by the 2..6 rows of the $[P^{-1}]$ matrix are the superposition of the inner differential modes for each transformer.

<figure><img src="image_5.png" /><figcaption>Figure 4: CM (left) and DM (right) magnetic flux path inside ICTs</figcaption></figure>

In order to highlight these differential modes, we perform electrical simulations of this converter structure with selected disturbances. As we want only one normal mode excited at one time, we generated a $\delta i_k$ current step on each branch with respect of the row of $[P^{-1}]$ linked to the selected mode. As an example, equation 10 shows the selected disturbance vector for the $\tau_2$ constant time.

$$[\delta i_k] = [-1 \ 2 \ -1 \ -1 \ 2 \ -1] \cdot \delta I \qquad (10)$$

The results of these electrical simulations confirm these analyses of normal modes with their linked time constants. The significant differences between these time constants make it difficult to control the converter. The aim of the control that we have implemented is to increase the dynamics of the slow modes. The direct model we have built can be depicted through the blocks diagram figure 5.

<figure><img src="image_6.png" /><figcaption>Figure 5: Direct model of the multicell inverter in orthogonal basis</figcaption></figure>
# Control toward currents balancing

## Inverse model

The control of the structure is derived from this direct model using the inverse model synthesis method. This method is a smart way to build the control scheme of the converter from the averaged model using symmetry. The quantities that we want to control are the output current $I_S$ and the magnetizing flux $\phi_k^{dif}$ (which are the differential modes ones) in the ICTs. These magnetic flux determine the use of the magnetic core, and we want to control that their mean values remain close to zero. Therefore, another basis change is defined by equation (11) in order to derive the differential mode magnetic flux $\phi_k^{dif}$ from the phase currents.

$$ \begin{pmatrix} I_S \\ k\phi_1^{dif} \\ \vdots \\ k\phi_{q-1}^{dif} \end{pmatrix} = [T] \cdot \begin{pmatrix} i_1 \\ i_2 \\ \vdots \\ i_q \end{pmatrix} \qquad (11) $$

The equation (12) shows the elements of the $[T]$ matrix in the q=6 case. The output current, which is the sum of the q phase currents $i_q$, and the magnetizing currents, which are the difference between the currents in two consecutive phases, can be computed from the phase currents using this matrix. The figure 6 represents the differential current $i_k^{dif}$ between phase $k+1$ and $k$, defined in equation 13. Because these differential currents are the quantity we want to control, the measurement of the phase currents themselves have to be done carefully.

$$ [T] = \begin{bmatrix} 1 & 1 & \cdots & \cdots & 1 \\ -1 & 1 & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & -1 & 1 \end{bmatrix} \qquad (12) $$

<figure><img src="image_7.png" /><figcaption>Figure 6: Differential current in InterCell Transformers</figcaption></figure>

$$ i_k^{dif} = i_{k+1} - i_k \qquad (13) $$

Eventually, the control scheme is shown in figure 7. One of the advantages of the current decoupling in the normal basis is shown on this scheme: indeed controller parameters are independent and can be set through the study of the system in this basis. We assume that the specifications of the converter operation can be held using PI controller [7]. The control loop ensures in the same time the current sharing balance and the dynamic control of the power output current. The PI controllers are tuned in a conventional manner, using the pole compensation method, in order to have a stable, fast and accurate control. Therefore, the closed loop dynamic is increased due to the proportional effect.
<figure><img src="image_8.png" /><figcaption>Figure 7: Control based on the inverse model of the multicell inverter</figcaption></figure>

## Experimental setup and results

### Prototype and Control implementation

In order to validate the control strategy, we developed and used the test vehicle shown in figure 8. This is a 300V, 1kW single phase inverter with $q = 6$ interleaved phases.

<figure><img src="image_9.png" /><figcaption>Figure 8: Prototype of a 6 cells interleaved inverter with ICTs</figcaption></figure>

The digital control is implemented using a DSP TMS320F28335 from Texas Instrument as illustrated in 9. Thanks to digital implementation, the control scheme has been improved by adding an integral anti-windup for PI controllers, and the dynamic limitation of the power transistor drivers. The chosen DSP is compliant with the computing power required to implement the balance control in real-time. This computing effort is estimated in the following table II in term of MAC (Multiply-Accumulate) cycles, taking into account the specificity of the sparse matrix.

<figure><img src="image_10.png" /><figcaption>Figure 9: Implementation of the balancing control on DSP</figcaption></figure>

## Results

Several tests have been carried out in order to validate the control performances. The first one is on the output current control loop: a step response of $I_s$ has been measured while switching the loads between two different values, and the output current dynamic was consistent with the controller settings. The second one is on the balancing of ICTs currents, and on the ability of the proposed control strategy
<table><caption>Table II: Computing effort, MAC cycles</caption><thead><tr><th>Operator</th><th>q=6</th><th>Generic case</th></tr></thead><tbody><tr><td><i>T</i></td><td>16 MAC</td><td><i>q</i> + 2(<i>q</i> − 1) = 3<i>q</i> − 2</td></tr><tr><td><i>T</i><sup>−1</sup> · <i>P</i><sup>−1</sup></td><td>36 MAC</td><td><i>q</i><sup>2</sup></td></tr><tr><td><i>P</i>.I. + anti-windup</td><td>30 MAC</td><td>5<i>q</i></td></tr><tr><td><i>P</i></td><td>28 MAC</td><td><i>q</i><sup>2</sup> (max)</td></tr><tr><td><i>Total</i></td><td>110 MAC</td><td>2<i>q</i><sup>2</sup> + 8<i>q</i> − 2 (max)</td></tr></tbody></table>

to respond on current unbalance. This is a more sensitive test, as we must be able to dynamically unbalanced the ICTs currents. Several solutions may be used, among which we choose one that consists to insert a small resistor in series with one phase of the ICTs. This resistor can be short-circuited by a MOS transistor, which allows a dynamic disturbance of the balance of currents. In steady state, the resistor is never short-circuited and we check that all the $i_k$ currents are identical, which validates the static balancing.

<figure><img src="image_11.png" /><figcaption>Figure 10: Closed loop response to a balance disturbance</figcaption></figure>

In order to study the dynamic response of the balancing control, the resistor is periodically shorted ($R = 0.476 \, \Omega$, $R_{shorted} = 0.045 \, \Omega$) and the difference between two ICTs currents ($i_1$ and $i_2$) is measured. The results are presented in 10, and show that the implemented dynamic balance control performs well. The response on closed loop is fast and show no static error, indeed the mean value of the currents difference is zero ($\langle |i_2 - i_1| \rangle = 0$). However, this solution is not an ideal one because several modes of the ICTs magnetization are excited. As it has been explained in the part on the converter direct model, the excitation of only one magnetization mode requires to apply a disturbance vector, as defined in equation 10, and therefore can be more easily tested using electrical simulation. The response of the inverter at such an excitation has been simulated and the performances of the control strategy have been validated.

## Conclusion

The performances of Multicell power converters with intercell magnetic coupling are of great interest regarding several criteria, as EMC performances or power integration. The design of an adapted and effective control of these kind of converters is an essential step to mastering the efficiency and reliability of their operation. The main goal of the control presented in this article is to ensure in an active way the balancing state, allowing the optimal functioning of the magnetic components. Furthermore, this control also act as the output current control loop. The approach used to design this control is based on the inversion of the direct model matrix. This direct model allows to highlight the power eigenmode, which is a common mode with fast time constant, and the magnetization eigenmodes, which are differential modes with slow time constants. This analysis through separate modes is in itself an interesting result of the proposed model. Synthesis of the regulation loop and calculation of controller parameters is based on the inversion of the direct model. A digital implementation of this controller has been achieved, and experimental results show that it performs well its functions regarding currents balancing and output current control.
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