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LMI for Controller Design of Multiple Solar PV Units Connected to Distribution Networks

Solar photovoltaic (PV) systems are a renewable energy source that can be integrated into existing power distribution networks for a clean and sustainable future. However, PV systems have stability and power quality issues due to, among other reasons, the strong dynamic interactions between PV units and the effects of atmospheric conditions. Thus, a controller needs to be designed to minimize stability issues and maximize power quality.

The System
The state-space representation:

$$ \begin{align} \dot{x}(t) &= Ax_i(t) + Bu_i(t)\\ y(t) &= Cx_i(t) \end{align} $$

where $$ x_i $$ is the state vector, $$ u_i $$ is the control signal for VSIs and $$ y_i $$ is the output vector of the $$ i^{th} $$ solar unit.

The Data
$$ \begin{align} x_i(t) = \begin{bmatrix} I_{di} & I_{qi} & V_{dci} \end{bmatrix}^{T}, A = \begin{bmatrix} -\frac{\sum R}{\sum L} & w & 0 \\ -w & -\frac{\sum R}{\sum L} & 0 \\ 0 & 0 & 0 \end{bmatrix}, B = \begin{bmatrix} \frac{V_{dci}}{\sum L} & 0 \\ 0 & \frac{V_{dci}}{\sum L} \\ -\frac{1}{C_i} & -\frac{1}{C_i} \end{bmatrix}, C = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}, u_i(t) = \begin{bmatrix} K_{di} & K_{qi} \end{bmatrix}^{T} \end{align} $$

where $$ I_{di}, I_{qi} $$ are d-and q-axis currents respectively; $$ V_{dci} $$ is the DC-bus voltage; $$ \sum R = R_{fi}+R_{Lg}+ \sum \beta \gamma R_{Lij} $$ is the resistance of the filter plus the resistance of the grid plus the resistance of the interconnecting line; $$ \sum L = L_{fi}+L_{Lg}+ \sum \beta \gamma L_{Lij} $$ is the inductance of the filter plus the inductance of the grid plus the inductance of the interconnecting line; $$ \beta $$ is the presence of current passing thru the interconnecting line due to other units; $$ \gamma $$ is the connectivity among various PV units; and $$ K_{di}, K_{qi} $$ are the d-and q-axis control inputs of the inverter respectively, where $$ K_{di} = m_{i}sin(\alpha_{i}), K_{qi} = m_{i}cos(\alpha_{i}) $$; $$ m_{i} $$ is the modulation index and $$ \alpha_{i} $$ is the firing angle.

The Optimization Problem
$$ \begin{align} u_i(t) = -Kx_i(t) \end{align} $$

The state feedback gain matrix K is:

$$ \begin{align} K = -R^{-1}B^{T}P \end{align} $$

K is optimized to minimize the performance index J:

$$ \begin{align} J = \int_{0}^{\infty}(x(t)^{T}Qx(t)+u(t)^{T}Ru(t)) \end{align} $$

The matrix P is obtained from the reduced-matrix algebraic Riccati equation:

$$ \begin{align} A^{T}P+PA-PBR^{-1}B^{T}P+Q<0 \end{align} $$

The LMI: LMI for Controller Design of Multiple Solar PV Units Connected to Distribution Networks
Using the Schur Complement: $$ P > 0, Q > 0, R > 0, $$ $$ \begin{align} \begin{bmatrix} A^{T}P+PA+Q & PB \\ B^{T}P & R \\ \end{bmatrix} < 0 \end{align}$$

Conclusion:
The calculated gain $$ K = -R^{-1}B^{T}P $$ is stable and related to the error of the current and DC-link voltage states which are expressed by $$ K1 $$ and $$ K2 $$ respectively, where $$ K1 = \begin{bmatrix} K11 & K12 \\ K21 & K22 \\ \end{bmatrix}, K2 = \begin{bmatrix} K13 \\ K23 \\ \end{bmatrix} $$

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/anaammostafiz/LMI-Solar-PV-Network