"4) Define circuit components and set their characteristic values\n",
"5) Assemble ciruit: Connect components and nodes, set system parameters: frequency, node list, component list\n",
"5) Assemble circuit: Connect components and nodes, set system parameters: frequency, node list, component list\n",
" \n",
"6) Set up the event which causes the switch to close. A switch event takes a value for the simulated time at which the switch state change\n",
" will occur, the affected switch and if it should open(False) or close(True) when the event triggers\n",
...
...
@@ -120,7 +120,7 @@
"\n",
"The lower equation describes the derivative of the current through all three components. After the discretization of both state equations it can be solved for the quantities $V_C(t+\\Delta t)$ and $I_{RLC}(t+\\Delta t)$ of the current time step. Inserting $V_C(t+\\Delta t)$ into the lower equation yields a term for $I_{RLC}(t+\\Delta t)$ which is only dependent on previous, known quantities $V_C(t)$, $I_{RLC}(t)$ and the previous voltage $V_{RLC}(t)$ across all three elements as well as $V_{RLC}(t+\\Delta t)$, the current voltage across all three elements.\n",
"\n",
"With this, the lower equation is a nodal equation which can be stamped into the system matrix and vectors. For this, the voltage $V_{RLC}(t+\\Delta t)$ has to be identified as node voltages, in this case $V_{RLC}(t+\\Delta t) = V_4$. After solving the system for a time step, the state $V_C(t+\\Delta t)$ gets updated via the upper state equation. This way, $V_C(t+\\Delta t)$, which will be $V_C(t)$ in the next time step, can then be used in the lower equation as a known term again."
"With this, the lower equation is a nodal equation which can be stamped into the system matrix and vectors. To do so, the voltage $V_{RLC}(t+\\Delta t)$ has to be identified as node voltages, in this case $V_{RLC}(t+\\Delta t) = V_4$. After solving the system for a time step, the state $V_C(t+\\Delta t)$ gets updated via the upper state equation. This way, $V_C(t+\\Delta t)$, which will be $V_C(t)$ in the next time step, can then be used in the lower equation as a known term again."
]
},
{
...
...
%% Cell type:markdown id: tags:
# Resistive Companion and State Space Nodal simulation approach using DPsim
%% Cell type:markdown id: tags:
## General Simulation Procedure
%% Cell type:markdown id: tags:
Using the MNA-based C++ solver [DPsim](https://www.fein-aachen.org/en/projects/dpsim/) for power system simulation with the dpsimpy module for python, the simulation is set up and carried out through the following steps:
1) Import modules
2) Define simulation parameters: name of the model, simulated time passed between two simulation steps, simulated time at which the simulation stops, time at which the switch closes through an event
3) Set up circuit nodes
4) Define circuit components and set their characteristic values
5) Assemble ciruit: Connect components and nodes, set system parameters: frequency, node list, component list
5) Assemble circuit: Connect components and nodes, set system parameters: frequency, node list, component list
6) Set up the event which causes the switch to close. A switch event takes a value for the simulated time at which the switch state change
will occur, the affected switch and if it should open(False) or close(True) when the event triggers
7) Create simulation instance, set parameters and start the simulation
8) Plot the results
%% Cell type:markdown id: tags:
### Resistive Companion approach: Sample circuit and MNA equations
Additionally to the RC equations above (which are identical to the SSN equations for the basic components R, L and C), the subsystem of a resistor, a capacitor and an inductor which are connected in series to the right of the switch can be merged into one combined component using the SSN approach. The state equations of the subcircuit are:
$\overrightarrow{\dot x} = \begin{bmatrix}
\dot V_C \\
\dot I_{RLC}
\end{bmatrix} =
\begin{bmatrix}
0 & \frac{1}{C} \\
-\frac{1}{L} & -\frac{R}{L}
\end{bmatrix}
\begin{bmatrix}
V_C \\
I_{RLC}
\end{bmatrix} +
\begin{bmatrix}
0 \\
\frac{1}{L}
\end{bmatrix} \cdot
V_{RLC}$
The lower equation describes the derivative of the current through all three components. After the discretization of both state equations it can be solved for the quantities $V_C(t+\Delta t)$ and $I_{RLC}(t+\Delta t)$ of the current time step. Inserting $V_C(t+\Delta t)$ into the lower equation yields a term for $I_{RLC}(t+\Delta t)$ which is only dependent on previous, known quantities $V_C(t)$, $I_{RLC}(t)$ and the previous voltage $V_{RLC}(t)$ across all three elements as well as $V_{RLC}(t+\Delta t)$, the current voltage across all three elements.
With this, the lower equation is a nodal equation which can be stamped into the system matrix and vectors. For this, the voltage $V_{RLC}(t+\Delta t)$ has to be identified as node voltages, in this case $V_{RLC}(t+\Delta t) = V_4$. After solving the system for a time step, the state $V_C(t+\Delta t)$ gets updated via the upper state equation. This way, $V_C(t+\Delta t)$, which will be $V_C(t)$ in the next time step, can then be used in the lower equation as a known term again.
With this, the lower equation is a nodal equation which can be stamped into the system matrix and vectors. To do so, the voltage $V_{RLC}(t+\Delta t)$ has to be identified as node voltages, in this case $V_{RLC}(t+\Delta t) = V_4$. After solving the system for a time step, the state $V_C(t+\Delta t)$ gets updated via the upper state equation. This way, $V_C(t+\Delta t)$, which will be $V_C(t)$ in the next time step, can then be used in the lower equation as a known term again.
#RLC-components: For the RC approach, the RLC-circuit simply consists of a resistor, an inductor and a capacitor.
#These individual components are RC models.
r_rlc=dpsimpy.emt.ph1.Resistor('r_rlc')
r_rlc.set_parameters(R=50)
l_rlc=dpsimpy.emt.ph1.Inductor('l_rlc')
l_rlc.set_parameters(L=0.01)
c_rlc=dpsimpy.emt.ph1.Capacitor('c_rlc')
c_rlc.set_parameters(C=0.000001)
# Assemble circuit
vs.connect([gnd,n1])
l1.connect([n2,n1])
l2.connect([n3,n2])
r1.connect([n3,n2])
c1.connect([gnd,n3])
sw.connect([n4,n3])
l_rlc.connect([n5,n4])
c_rlc.connect([n6,n5])
r_rlc.connect([gnd,n6])
system=dpsimpy.SystemTopology(60,[gnd,n1,n2,n3,n4,n5,n6],[vs,l1,l2,r1,c1,sw,r_rlc,l_rlc,c_rlc])#Parameters are: (frequency=60Hz, system nodes, system components)
#Logging
logger=dpsimpy.Logger(model_name)
logger.log_attribute('v3','v',n3)#log voltage at node 3
logger.log_attribute('i_rlc','i_intf',l_rlc)#log current through RLC branch, inductor current arbitrarily chosen here (equal to resistor and capacitor current)
logger.log_attribute('i_c1','i_intf',c1)#log current through capacitor c1
#Events
sw_event=dpsimpy.event.SwitchEvent(switch_close_time,sw,True)#Define event which changes the switch state
#Parameters:
#(Simulated time value for which the event activates, affected switch, open[False] or close[True] the switch)
#RLC components: In the SSN approach, more complex circuits are combined into a single component.
#Instead of a resistor, an inductor and a capacitor a single SSN RLC element is used.
rlc=dpsimpy.emt.ph1.Full_Serial_RLC('rlc')
rlc.set_parameters(R=50,L=0.01,C=0.000001)
# Assemble circuit
vs.connect([gnd,n1])
l1.connect([n2,n1])
l2.connect([n3,n2])
r1.connect([n3,n2])
c1.connect([gnd,n3])
sw.connect([n4,n3])
rlc.connect([gnd,n4])
system=dpsimpy.SystemTopology(60,[gnd,n1,n2,n3,n4],[vs,l1,l2,r1,c1,sw,rlc])#Parameters are: (frequency=60Hz, system nodes, system components)
#Logging
loggerSSN=dpsimpy.Logger(model_name_SSN)
loggerSSN.log_attribute('v3','v',n3)#log voltage at node 3
loggerSSN.log_attribute('i_rlc','i_intf',rlc)#log current through RLC branch, inductor current arbitrarily chosen here (equal to resistor and capacitor current)
loggerSSN.log_attribute('i_c1','i_intf',c1)#log current through capacitor c1
#Events
sw_event=dpsimpy.event.SwitchEvent(switch_close_time,sw,True)#Define event which changes the switch state
#Parameters:
#(Simulated time value for which the event activates, affected switch, open[False] or close[True] the switch)
Marvin Tollnitsch <marvin.tollnitsch@rwth-aachen.de>