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I am trying to obtain the state space model of the DC microgrid shown in the figure. It is made of three converters connected in parallel. Each one is modeled with a voltage source that outputs the nominal voltage \$V_{ref}\$, a CCVS that models a droop control, and an RLC network that models the dynamics of the output voltage caused by load changes. The differential equations of one converter (for example, nº 1) are the following:

$$C_1\frac{dv_{c1}}{dt} + I_{o1} = i_{l1} +\frac{(V_{ref} - K_1I_{o1} - v_{c1})}{R_1}$$ $$L_1\frac{di_{l1}}{dt} + v_{c} = V_{ref} - K_1I_{o1} $$ From there, we can write the state space representation of one converter as a subsystem:

$$\begin{pmatrix} \frac{dv_{c1}}{dt} \\ \frac{di_{l1}}{dt} \end{pmatrix} = \begin{pmatrix} -\frac{1}{C_1R_1} & \frac{1}{C_1} \\ -\frac{1}{L_1} & 0 \end{pmatrix} \begin{pmatrix} v_{c1}\\ i_{l1} \end{pmatrix} + \begin{pmatrix} \frac{1}{R_1C_1} & -\frac{K_1}{R_1C_1} - \frac{1}{C_1} \\ \frac{1}{L_1} & -\frac{K_1}{L_1} \end{pmatrix} \begin{pmatrix} V_{ref} \\ I_{o1} \end{pmatrix} $$

Now the issue is that we don't know the output currents of each converter \$I_{oi}\$, but we know the total load \$I_L\$, and we want to write the state space representation of the entire system, where the input variables are just \$I_L\$ and \$ V_{ref}\$. Is there any way to do that by combining the state space models of every converter? The connection in parallel implies the following restrictions:

$$I_{o1} + I_{o2} + I_{o3} = I_L$$ $$v_{c1} = v_{c2} = v_{c3}$$

The total load is equal to the sum of the output currents, and the capacitor voltages are the same for every converter.

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FernandoPT
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  • For a DC microgrid, wouldn't the cabling resistance play an important role? We could say that the three generators are in parallel and the load is such-and-such, but in reality the cabling resistance would have a fairly large impact on results for everything except a bench-top grid. – rdtsc Oct 25 '22 at 12:02
  • It does, and also component tolerances that would lead the outputs to being not quite the same, but this looks more like an ideal exersice – Voltage Spike Oct 25 '22 at 17:19

1 Answers1

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I don't know but the matrix would look something like this:

$$\begin{pmatrix} \frac{dv_{c1}}{dt} \\ \frac{di_{l1}}{dt} \\ \frac{dv_{c2}}{dt} \\ \frac{di_{l2}}{dt} \\ \frac{dv_{c3}}{dt} \\ \frac{di_{l3}}{dt} \\ 0\\ 0\\ I_L \end{pmatrix} = \begin{pmatrix} -\frac{1}{C_1R_1} & \frac{1}{C_1} & 0 & 0 & 0 & 0\\ -\frac{1}{L_1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{C_2R_2} & \frac{1}{C_2} & 0 & 0\\ 0 & 0 &-\frac{1}{L_2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{1}{C_3R_3} & \frac{1}{C_3} \\ 0 & 0 & 0 & 0 & -\frac{1}{L_3} & 0 \\ -1 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & -1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} v_{c1}\\ i_{l1} \\ v_{c2}\\ i_{l2}\\ v_{c3}\\ i_{l3}\\ 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} \frac{1}{R_1C_1} & -\frac{K_1}{R_1C_1} - \frac{1}{C_1} & 0 & 0 & 0 & 0 \\ \frac{1}{L_1} & -\frac{K_1}{L_1} & 0 & 0 & 0 & 0 \\ 0 & 0 &\frac{1}{R_2C_2} & -\frac{K_2}{R_2C_2} - \frac{1}{C_2} & 0 & 0 \\ 0 & 0 & \frac{1}{L_2} & -\frac{K_2}{L_2} & 0 & 0 \\ 0 & 0 & 0 & 0 &\frac{1}{R_3C_3} & -\frac{K_3}{R_3C_3} - \frac{1}{C_3} \\ 0 & 0 & 0 & 0 &\frac{1}{L_3} & -\frac{K_3}{L_3} \end{pmatrix} \begin{pmatrix} V_{ref} \\ I_{o1} \\ V_{ref} \\ I_{o2} \\ V_{ref} \\ I_{o3} \\ \end{pmatrix} $$

The matrix/system of equations is overdetermined so you would need to find the tools to solve it . You may need to modify the circuit to overcome that problem, maybe try adding some series resistance in each stage

Voltage Spike
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