# LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Controller Design of Multiple Solar PV Units Connected to Distribution Networks

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:

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax_{i}(t)+Bu_{i}(t)\\y(t)&=Cx_{i}(t)\end{aligned}}}

where ${\displaystyle x_{i}}$ is the state vector, ${\displaystyle u_{i}}$ is the control signal for VSIs and ${\displaystyle y_{i}}$ is the output vector of the ${\displaystyle i^{th}}$ solar unit.

## The Data

{\displaystyle {\begin{aligned}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{aligned}}}

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

## The Optimization Problem

{\displaystyle {\begin{aligned}u_{i}(t)=-Kx_{i}(t)\end{aligned}}}

The state feedback gain matrix K is:

{\displaystyle {\begin{aligned}K=-R^{-1}B^{T}P\end{aligned}}}

K is optimized to minimize the performance index J:

{\displaystyle {\begin{aligned}J=\int _{0}^{\infty }(x(t)^{T}Qx(t)+u(t)^{T}Ru(t))\end{aligned}}}

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

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

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

Using the Schur Complement: ${\displaystyle P>0,Q>0,R>0,}$ {\displaystyle {\begin{aligned}{\begin{bmatrix}A^{T}P+PA+Q&PB\\B^{T}P&R\\\end{bmatrix}}<0\end{aligned}}}

## Conclusion:

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

## Implementation

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