# Decentralized Robust Control of a Network of Inverter-Based Distributed Generation Systems
## 2 Methodology
### Dynamical Equations of a Network of Inverters  
The electrical model of each of the inverters in this paper is illustrated in Figure 2, in which the output is connected to an LC filter and the input is connected to an energy storage system. The purpose of having LC filters is to eliminate the total harmonic distortion of the inverter output current. These passive filters are employed to curtail the amplitude of the harmonics and thereby improve the power quality [31]. In spite of the benefits of passive filters, they bring about undesired resonances, rendering the control design process more challenging [32]. Decentralized controllers aim to minimize the error between the load voltage \(v_{load_{i}}\) and a sinusoidal reference voltage \(v_{ref_{i}}\) such that the entire network is robust against the coupling strength among the inverters, parameter uncertainties of the LC filter, and the undesired disturbances coming from the load variations.  
The associated dynamical equations of inverter \(\#i\) (\(i=1,...,N\)) in Figure 2 are obtained as follows:  
\[\begin{array}{l}i_{inv_{1}}=G_{inv_{1}}(v_{inv_{1}}-v_{load_{i}}),\ G_{inv_{ 1}}:=\frac{1}{L_{i8}}\\ v_{load_{i}}=G_{v_{i}}[i_{inv_{2}}-i_{load_{i}}-\sum\limits_{k\in\mathcal{N} }(i)\underbrace{s_{ik}G_{ik}(v_{load_{i}}-v_{load_{k}})}_{i_{ik}}]\\ G_{v_{2i}}:=\frac{1}{C_{i8}},\ G_{ik}:=\frac{1}{L_{i8}+R_{ik}}\end{array} \tag{3}\]  
where \(L_{i}\) and \(C_{i}\) represent the inductance and capacitance of the LC filter of inverter \(\#i\) and \(L_{ik}\) and \(R_{ik}\) represent the inductance and resistance between inverters \(\#i\) and \(\#k\) (\(i=1,...,N\) and \(k\in\mathcal{N}(i)\)). Furthermore, \(s_{ik}\) in this equation comes from Equation (1), and determines the sign of each term.  
In order to formulate the state space model of inverter \(\#i\), the following state space variables are defined: \(x_{i}=[i_{inv_{1}}\ v_{load_{i}}]^{T}\) is the state vector, \(u_{i}\) is the \(i^{th}\) control input, \(d_{i}\) is the \(i^{th}\) disturbance input (load current), and \(x_{ik}=i_{ik}\) is the interconnection current between inverters \(\#i\) and \(\#k\). Therefore, the state space equations corresponding to Equation (3) are obtained as follows:  
\[\begin{array}{l}\dot{x}_{i}=A_{i}x_{i}+B_{i}u_{i}+D_{i}d_{i}+\sum\limits_{k \in\mathcal{N}(i)}A_{ik}x_{ik}\\ y=C_{i}x_{i}\end{array} \tag{4}\]  
where \(A_{i}\), \(B_{i}\), \(D_{i}\), \(A_{ik}\), and \(C_{i}\) are represented as follows:  
\[\begin{array}{l}A_{i}=\left[\begin{array}{cc}0&-\frac{1}{L_{i}}\\ \frac{1}{C_{i}}&0\end{array}\right],\ B_{i}=\left[\begin{array}{c}\frac{1}{L _{i}}\\ 0\end{array}\right],\ D_{i}=\left[\begin{array}{c}0\\ -\frac{1}{C_{i}}\end{array}\right]\\ A_{ik}=\left[\begin{array}{c}0\\ -\frac{1}{C_{i}}\end{array}\right],\ C_{i}=\left[0\ \ 1\right]\end{array}\]  
Figure 2: The electrical circuit of inverter \(\#i\) (\(i=1,...,N\)) with an LC filter, showing its connection to inverter \(\#k\) (\(k\in\mathcal{N}(i)\)).  
The closed-loop block diagram of inverter #\(i\) is obtained according to Equation (3), and is illustrated in Figure 3, where \(K_{i}(s)\) and \(M_{i}\) represent the decentralized controller and inverter gain for inverter #\(i\), respectively. Moreover, \(u_{i}\) is the command signal generated by the controller \(K_{i}(s)\). In this block diagram, \(i_{load_{i}}\) is considered to be an external disturbance to the system.  
Thus far, the dynamical model of inverter #\(i\) consisting of the LC filter and interconnection line dynamics has been formulated. It can be observed that the interconnection line dynamics \(G_{ik}\) in (3) are shared between the neighboring inverters. In the next section, the state space representation of the entire network of IBDGs is obtained such that the shared dynamics (interconnection line dynamics) are considered for both neighboring inverters #\(i\) and #\(k\).