## 3 Proposed Non-Linear Current Control Scheme of Grid Connected Inverter with LCL Filter
### Output Current Control
#### 3.1.1 Proposed Active Current Control  
To drive \(i_{gd}\), as an active current, into \(i_{gd}^{ref}\), \(e_{5}\) has to be forced to zero through proper selection of control variables. To do so, the term \(k_{1}e_{5}\) is added or subtracted to or from the right-hand side of \(de_{5}/dt\) in (24), as shown in (26). \[\frac{de_{6}}{dt}=\omega e_{5}+\frac{e_{4}}{L_{2}} \tag{26}\]  
By defining the new variable of \(Z_{1}\) as (27), (26) is rewritten as (28)  
\[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]  
\[\frac{de_{5}}{dt}=-k_{1}e_{5} \tag{28}\]  
The derivative of \(Z_{1}\) is named \(V_{1}\) in the rest of the paper, as given in (29). \[\frac{de_{5}}{dt}=-k_{1}e_{5} \tag{29}\]  
Substituting \(de_{5}/dt\) and \(de_{6}/dt\) from (24)-(25) into (29), and defining the intermediate variable of \(u_{1}=de_{3}/dt\), (29) is rewritten as:  
\[\frac{dZ_{1}}{dt}=V_{1}=-\omega^{2}e_{5}-\omega\frac{e_{4}}{L_{2}}+\frac{u_{1 }}{L_{2}}-k_{1}\omega e_{6}+k_{1}\frac{e_{3}}{L_{2}} \tag{30}\]  
If \(Z_{1}\) is enforced to zero, based on (28), \(de_{5}/dt\) will be equal to \(-k_{1}e_{5}\); and then, for a positive value of \(k_{1}\), \(e_{5}\) will go to zero exponentially. To do so, the positive definite Lyapunov function \(V_{h1}\) is defined as follows:  
\[V_{h1}=\frac{1}{2}e_{5}^{2}+\frac{1}{2}Z_{1}^{2} \tag{31}\]  
The derivative of (31) is given in (32). \[\frac{dV_{h1}}{dt}=e_{5}\frac{de_{5}}{dt}+Z_{1}\frac{dZ_{1}}{dt} \tag{32}\]Substituting (28) and (30) into (32) results in (33). \[\frac{dV_{h1}}{dt}=-k_{1}e_{5}^{2}+Z_{1}e_{5}+Z_{1}V_{1} \tag{33}\]  
Based on (33), if \(V_{1}\) is forced to (34), (33) is derived to (35). \[\frac{dV_{h1}}{dt}=-k_{1}e_{5}^{2}+Z_{1}e_{5}+Z_{1}V_{1} \tag{34}\]  
\[\frac{dV_{h1}}{dt}=-k_{1}e_{5}^{2}+Z_{1}e_{5}+Z_{1}V_{1} \tag{35}\]  
Accordingly, selecting the positive values for \(k_{1}\) and \(k_{2}\) guarantees the force of \(e_{5}\) and \(Z_{1}\) to zero based on the Lyapunov stability theorem [34]. Now, V\({}_{1}^{desired}\) in (34) is added and subtracted to the right-hand side of (30) as:  
\[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]0  
Now, the new variable of \(Z_{2}\) is defined as given in (37). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]1  
Using the expression of \(Z_{2}\) from (37), (36) changes to (38). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]2  
The derivative of \(Z_{2}\) is also as given in (39). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]3  
From the above discussions, it is concluded that in order to drive \(e_{5}\) to zero, \(Z_{1}\) and \(Z_{2}\) should also move to zero. For this purpose, another positive definite Lyapunov function \(V_{h2}\) is defined as shown in (40). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]4  
The derivative of the Lyapunov function \(V_{h2}\) with respect to the time is as given in (41). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]5  
substituting \(de_{5}/dt\) and \(dZ_{1}/dt\) respectively from (28) and (38), (41) is changed to (42). \[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]6  
Therefore, if \(dZ_{2}/dt\) is equal to \(-Z_{1}-k_{3}Z_{2}\), then, based on (39), is achieved by selecting \(n_{1}\) as (43)  
\[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]7  
Equation (42) is equal to (44)  
\[Z_{1}=-\omega e_{6}+\frac{e_{3}}{L_{2}}+k_{1}e_{5} \tag{27}\]8Therefore, according to (44) and based on the Lyapunov theorem, all variables of \(e_{5}\), \(Z_{1}\), and \(Z_{2}\) move exponentially to zero. Deriving \(e_{5}\) to zero exponentially indicates the independent tracking of the reference value of \(i_{gd}^{ref}\) from other variables, especially without any interaction with the reactive current dynamics. The resonance modes are also dampened. Moreover, the poles of the control system, related to the active current \(\left(i_{gd}\right)\) control channel, are equal to \(k_{1}\), \(k_{2}\), and \(k_{3}\), which are the control parameters in (43).