1. Introduction

2. The proposed control design

2.2. First Step of the Design

Let now address the problem of designing ${\mathbf{C}}_v\left(s\right)$. Since the active power control loops are decoupled, it is possible to consider only the transfer functions corresponding to the pairings $(i_{qrefi},v_i)$, see Figure 1. To this scope the matrix plant ${\mathbf{P}}_n\left(s\right)$ is written by evidencing the $2\ell$ inputs and outputs as follows

The square matrix plant with inputs $i_{qrefi}$ and outputs $v_i$ used in this step is then given by.

$$\begin{array}{ccc}\hfill {\mathbf{P}}_n^v\left(s\right)& =& \left(\begin{array}{cccccc}{P}_{22}& P_{24}& P_{26}& \dots & \dots & P_{2(2\ell )}\\ P_{42}& P_{44}& P_{46}& \dots & \dots & P_{4(2\ell )}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ P_{(2\ell -2)2}& P_{(2\ell -2)4}& P_{(2\ell -2)6}& \dots & \dots & P_{(2\ell -2)(2\ell )}\\ P_{(2\ell )2}& P_{(2\ell )4}& P_{(2\ell )6}& \dots & \dots & P_{(2\ell )(2\ell )}\end{array}\right)\hfill \end{array}$$

Employing the method of the Effective Open-Loop Transfer Function (EOTF), $P_i^{eotf}\left(s\right)$, the design of the IMC controller $C_{vi}^{imc}\left(s\right)$ is developed for model $P_i^{eotf}\left(s\right)$ rather than ${\mathbf{P}}_n^v\left(s\right)$, thus avoiding a more complex MIMO design.

The expression of $P_i^{eotf}\left(s\right)$ is given by [28]

$$P_i^{eotf}\left(s\right)=\frac{{\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}}{DRG{A}_{ii}\left(s\right)}$$

(6)

where ${\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}$ and $DRG{A}_{ii}\left(s\right)$ denote, respectively, the i-th diagonal element of ${\mathbf{P}}_n^v\left(s\right)$ and of the Dynamic Relative Gain Array (DRGA) calculated by

$$DRG{A}_{ii}\left(s\right)={[{\mathbf{P}}_n^v\left(s\right)\otimes {\left({\mathbf{P}}_n^v{\left(s\right)}^{-1}\right)}^T]}_{ii}$$

The IMC voltage control matrix is

$${\mathbf{C}}_v^{imc}\left(s\right)=\mathrm{diag}\left\{\underset{C_{v1}^{imc}\left(s\right)}{\underbrace{\frac{1}{P_1^{eotf}\left(s\right){(1+s{\lambda }_{v1})}^{q_{v1}}}}}\dots \dots \underset{C_{v\ell }^{imc}\left(s\right)}{\underbrace{\frac{1}{P_{\ell }^{eotf}\left(s\right){(1+s{\lambda }_{v\ell })}^{q_{v\ell }}}}}\right\}$$

where $q_{v1},\dots q_{v\ell }$ are positive integers chosen to make ${\mathbf{C}}_v^{imc}\left(s\right)$ realizable.

From ${\mathbf{C}}_v^{imc}\left(s\right)$ it is obtains the feedback control matrix

$${\mathbf{C}}_v\left(s\right)=\mathrm{diag}\left\{\frac{1}{s{\lambda }_{v1}{P}_1^{eotf}\left(s\right)}\dots \dots \frac{1}{s{\lambda }_{v\ell }{P}_{\ell }^{eotf}\left(s\right)}\right\}$$

(7)

The adjustable control parameters ${\lambda }_{v1},\dots {\lambda }_{v\ell }$ are selected with the objective of reducing the external interactions. To this scope we formulate the controller design problem into an optimization framework. In details, ${\lambda }_{v1},\dots {\lambda }_{v\ell }$ are obtained by solving the following problem

$$\underset{{\lambda }_{v1},\dots {\lambda }_{v\ell }}{min}\left|\right|{\mathbf{W}}_d(j\omega ){\left|\right|}_{\infty }\omega \in {\mathsf{\Omega }}_v$$

(8)

subject to

$${\lambda }_{vi}^m\le {\lambda }_{vi}\le {\lambda }_{vi}^M$$

where ${\mathsf{\Omega }}_v$ is a a frequency range of interest and ${\mathbf{W}}_d(j\omega )$ is the interaction matrix obtained from the following closed voltage control matrix

$${\mathbf{W}}_v(j\omega )={\left({\mathbf{I}}_{\ell \times \ell }+{\mathbf{P}}_n^v(j\omega ){\mathbf{C}}_v(j\omega )\right)}^{-1}{\mathbf{P}}_n^v(j\omega ){\mathbf{C}}_v(j\omega )$$

by setting ${\left[{\mathbf{W}}_v(j\omega )\right]}_{ii}=0$. In this way the response $v_i$ due to a change of $v_{desj}$ ($j\ne i$) is effectively attenuated.

Remark 1.

As above described, the design of ${\mathbf{C}}_v\left(s\right)$ uses model in (6). The EOTF relates $i_{qrefi}$ to $v_i$ when the i-th loop is open while all other loops are closed under the condition of perfect control. To explain this concept, let consider the actual relationship between $i_{qrefi}$ to $v_i$ given by [28]

$$v_i=\left[{\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}-{\mathbf{p}}^{ir}\left(s\right){\tilde{\mathbf{C}}}_{vi}\left(s\right){\left({\mathbf{I}}_{(\ell -1)\times (\ell -1)}+{\mathbf{P}}_n^{vi}\left(s\right){\tilde{\mathbf{C}}}_{iv}\left(s\right)\right)}^{-1}{p}^{ic}\left(s\right)\right]i_{qrefi}$$

(9)

where ${\tilde{\mathbf{C}}}_{vi}=\mathrm{diag}\{C_{v1},C_{v2},\dots C_{v(i-1)},C_{v(i+1)},\dots C_{v\ell }\}$, ${\mathbf{P}}_n^{vi}\left(s\right)$ denotes a transfer functions matrix where both the i-th row and column are removed from ${\mathbf{P}}_n^v\left(s\right)$, ${\mathbf{p}}^{ir}\left(s\right)$ and ${\mathbf{p}}^{ic}\left(s\right)$ are the i-th row and column vector of matrix ${\mathbf{P}}_n^v\left(s\right)$ where ${\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}$ is discarded, respectively.

If the following approximation

$${\mathbf{H}}_i(j\omega )={\mathbf{P}}_n^{vi}(j\omega ){\tilde{\mathbf{C}}}_{vi}(j\omega ){\left({\mathbf{I}}_{(\ell -1)\times (\ell -1)}+{\mathbf{P}}_n^{vi}(j\omega ){\tilde{\mathbf{C}}}_{vi}(j\omega )\right)}^{-1}\simeq {\mathbf{I}}_{(\ell -1)\times (\ell -1)}$$

(10)

holds in the frequencies range smaller than the cross-over frequency ${\omega }_{ci}$, then eq. (9) can be reasonably simplified as follows

$$v_i=\left({\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}-{\mathbf{p}}^{ir}\left(s\right){\left({\mathbf{P}}_n^{vi}\right)}^{-1}\left(s\right){\mathbf{p}}^{ic}\right)i_{iqrefi}=\frac{{\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{ii}}{DRG{A}_{ii}\left(s\right)}{i}_{iqrefi}=P_i^{eotf}\left(s\right)i_{iqrefi}$$

Condition (10) guarantees the perfect control and it is fulfilled if $C_{vj}\left(s\right)$, ($j=1,\dots \ell$, $j\ne i$) has a pole in $s=0$ and its gain is greater than that of ${\left[{\mathbf{P}}_n^v\left(s\right)\right]}_{jj}$, respectively.

3. Cases Study and Simulation Results

4. Conclusions

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