State-Space | Example

SIMPLE SINGLE-PHASE EXAMPLE
This simple example compares the simulation of an actual circuit to its equivalent state-space model. The circuit drawing is shown in Figure 7. The design file is simpleTest_state_space.ecf.
It has one independent state variable and two external connectivity pins (nodes). The current and voltage vectors have a size of 2. The currents i1 and i2 are entering the circuit pins as shown on the diagram. Along the real network, the state space representation is tested using the two methods to import input data (data file and manually entered matrices). The voltage sources have been set to 600V with a 60 Hz frequency. One-phase node connectors have been used to connect bundle pins to buses.

The symbolic matrix A is given by:

EQ 09

The other matrices are:

EQ 10

EQ 11

EQ 12

EQ 13

Since both state-space and circuit representations are mathematically identical, the simulation waveforms from both circuits should be the same. During the simulation, the breaker is opened at t=30ms and then closed at t=60ms. In Figure 10, the simulation results (v2 voltage) validate the state space for both input methods.

EXAMPLE OF A THREE-PHASE NETWORK
The test case is a reduction of a small 225kV network as presented in Figure 9.

The network equivalent has two terminals BUS1 and BUS3. Only the two central PI-lines will be kept. A 10 Ohm single phase fault (phase A) is applied between these PI-lines at t=30ms and is
eliminated at t=60ms. This scenario is used for both networks (reduced one and complete one).

The network reduction has been done within EMTP to get the 25 poles input data file. Figure 10 below shows the reduced network.

Voltage sources have been replaced by a Norton equivalent to cope with the State Space modelling which is considered as an impedance. Simulation results of the fault current and voltage node are superposed for the complete network (HYP1) and the reduced one (HYP2).

Fault node voltage (phase A)

The superposition of the results is quite good. It demonstrates the State-Space representation relevance where precision is equally the same as the real network (relative difference).

Below the data used for each cases:

• 50Hz frequency
• RL load on BUS3 R=480 Ohm, L=3.06H
• Coupling PI-Line between AC1 and AC2 voltage source R=17.5 Ohm, L=351mH, C=189µF
• Other PI-Lines R=1.75 oHm, L=35.1 mH, C=189 µF
• BUS1 and BUS3 resistance R=1 MOhm
• The two Thevenin equivalent voltage sources (AC1 and AC2) are initialized from Load Flow. Constraints are a swing bus on AC1 which set the voltage as 240kV RMS
• (225kV is the reference voltage) A PQ constraint is set for AC2 voltage source P=-200 MW and Q=-10 MVar. The source RL values are R=4 Ohm, L=70 mH
• Single phase fault applied at t=30ms and eliminated at t=60ms
• Fault resistance value R=10 Ohm

Then, for the reduced network (Figure 10) the two Norton equivalents have also been initialized from EMTP Load Flow. Values of each current (balanced three-phase system) source are:

• AC2 5.562e3 for module peak and -90.2 deg for angle
• AC1 6.923e3 for module peak and -74.8 deg for angle

REFERENCES
[1] B. Bruned, C. Martin, S. Dennetière, Y. Vernay, "Implementation of a unified modelling between EMT tools for network Studies", submitted to the International Conference on Power Systems Transients (IPST2017) in Seoul, Republic of Korea, 2017.