In this post, we will analyze the circuit below for the synthetic testing of Circuit Breakers (CBs) in the case when CB is open and close. The idea is to solve the electrical transient mathematically but the following approach will be based on a simulation analysis of synthetic testing.
According to , synthetic tests are based on a two part test that is done all at once. “The test is performed by combining a moderate voltage source which supplies the full primary short circuit current with a second, high voltage, low current, power source which injects a high frequency, high voltage pulse at a precise time near the natural current zero of the primary high current. The purpose of synthetic tests is to reproduce conditions that closely simulate those that prevail in the interrupter during the high current arcing and the high voltage recovery periods” .
There are a number of synthetic test schemes that have been developed, but in practice they are all only a variation of the basic voltage or current injection schemes :
1. Current Injection Method
This method is characterized by the injection of a pulse of current that is supplied by the high voltage source.
The two major current injection methods are:
- Parallel Current Injection Circuit
Series Current Injection Circuit
2. Voltage Injection Method
In practice, all testing laboratories more frequently use the parallel current injection technique test (Figure 1)  , described in detail below:
The high current source is composed of:
- A short circuit generator,
- A back-up CB for the protection of the test generator,
- A set of current limiting reactors,
- A making switch and,
- An Isolation Breaker (IB) that isolates the current circuit from the high voltage circuit.
The high voltage section of the circuit is made up of:
- A high voltage source consisting of a capacitor bank that is charged to a predetermined high voltage level. Connected in series with the capacitor is one side of a triggered spark gap (TG).
- The other side of the trigger gap is connected to a group of frequency tuning reactors.
- Connected in series with these reactors there is a short line fault (SLF) TRV shaping network, which consists of a combination of capacitors and reactors that in most instances are connected in a classical pi (π) circuit configuration.
The schematic diagram of the parallel current injection shown in Figure 1 is presented again in a more simplified manner in Figure 6 (in a red rectangle the circuit part of our initial problem). The test is initiated by closing the making switch (MS), which initiates the flow of the current i1, from the high current source through the IB and the Test Breaker (TB). As the current approaches its zero crossing the spark gap is triggered and at time t1 (see Figure 2) the injected current i2 begins to flow. The current i1 + i2 flows through the TB until the time t2 is reached. This is the time when the main current i1 goes to zero and when the IB separates the two power sources. At time t3 the injected current is interrupted and the high voltage supplied by the high voltage source provides the desired TRV which subsequently appears across the terminals of the CB that is being tested.
IEC Standard of Transient Recovery Voltage (TRV) Envelops
Short circuit test requires circuit with response specified by IEC Standards  in order to control the TRV. The TRV produced from tests must follow the specification as required by the standards.
Case of Two Parameters 
Case of Four Parameters 
Based on what we want to solve and what we explain above related to synthetic testing of CBs, below is the developed parallel current injection model in MATLAB/Simulink (the files can be emailed upon request):
Based on ANSI/IEEE Std. C37  parallel current injection circuit operation can be described as follows:
The test is initiated by closing the making switch which allows current i1 to flow through the current limiting reactor Lc, the auxiliary CB, and the TB.
Prior to the interaction interval, the spark gap is triggered at time t1, (see Figure 1) and current i2 is injected and flows through the high-voltage reactor Lv, and TB.
The current through the TB is i1 plus i2 until time t2 when the auxiliary breaker isolates the current circuit from the voltage circuit.
The current through the TB is i2 until time t3, when the TB interrupts and the transient recovery voltage (TRV) is impressed across the TB.
Figure 9 shows the voltage across the TB before and after the interruption. As shown the arc voltage was appearing across the TB during the separation of the contacts. Then, when it comes to current zero, the TB totally opens the contacts with very small amount of current flowing through the TB.
Figure 10 shows the short circuit current supplied by the generator forced to the TB. Short circuit current waveform was affected by the DC and AC components in the test circuit.
A synthetic test is also about to inject the current pulse at a few microseconds before the short circuit current reaches the zero point in order to provide restriking voltage across the TB. Therefore, the capacitor bank has to be fully charged before it is used to supply the injection current. Figure 11 shows the charging and discharging process of the capacitor bank.
Figure 12 shows the current triggered from the capacitor bank in order to provide the restriking voltage to TB after the auxiliary breaker isolate the short circuit current from the TB. The magnitude of the injected current should be adjusted so that the rate of change of the injected current (di/dt) and the rate of change of the corresponding rated power frequency current (di/dt)p are equal at their respective current zeros. The timing for initiation of the current pulse is controlled so that the time during which the arc is fed only by injected, current is no more than one-quarter of the period of the injected frequency .
Total current that flows through the TB during the synthetic test is shown in Figure 13. The most important part of this simulation result is the effect of injection current as shown in Figure 14. It is recommended  that the frequency for the injected current to be kept within the range of 300 to 1000 Hz. The limits depend primarily on the characteristics of the arc voltage.
The final analysis for the synthetic simulation is to show the unsuccessful interruption result of the TB. The failure to recover the voltage means the TB has failed to perform as fault clearer. Figure 15 shows that the TB fails to isolate the fault because it allows the restriking of short circuit current, where the short circuit current continues to flow through the TB after the interruption.
Results obtained from the synthetic test simulation were conducted using the current injection method. In principle, the current injection method and the voltage injection method are the same. As mentioned before, the only difference is that the output of high voltage source is injected across the open contacts of the TB following the interruption of the short circuit current. It is recommended that both test methods are modeled to perform the simulation. However voltage injection method is not too popular as a testing method because it requires a very accurate timing for the voltage injection.
Basically in actual synthetic test, the main disadvantage is that these tests are primarily a single loop test where it is still very difficult to do a fast reclosing with extended arcing times. Another disadvantage is that this method is not suitable for testing interrupters which have impedance connected in parallel with the interrupter contacts in which case it is likely that the full recovery voltage cannot be attained due to the power limitations of the high voltage source .
For failure interruption simulation, the test circuit breaker is modeled to be reclosed after the interruption to show that the breaker fails to clear the fault. The arc fails to extinguish during the contacts separation whereas the breaker allows the short circuit current to continue flowing.
Appendix: Explanations about the elements of the developed model
As mentioned, synthetic test has two main sections, the current source and the voltage source. The current source is a short circuit current supply with possible maximum current. It has high current flow at low voltage. For the voltage source it is high voltage supply and has low current flow. In real synthetic test, voltage source is a fully charged capacitor bank before it can be applied for testing. For the purposes of a simulation, this capacitor can be modeled as a dc voltage source. The spark gap can be modeled as a controlled time switch, because in real synthetic test, the spark was triggered with special controlled circuit. The additional circuit is to control spark triggered at the desired moment and slightly before the short circuit current reaches it natural zero.
Short circuit generator
Purpose: to provide the energy used during the test (high current through the TB in a short duration). For the simulation, the generator was modeled as AC supply with RL impedance.
Notes: The transient and sub-transient reactances must be as small as possible to provide the maximum short circuit current (leakage flux must be minimal and individual windings as close coupled as possible) .
Master Circuit Breaker
Auxiliary Circuit Breaker
Purpose: to apply short circuit current at the desired moment during the test. This switch is used to connect the generator to the test circuit at a precise point on the voltage wave and high speed operation is essential to enable the precision in closing instant to be obtained. Pre-arcing during the closure of the switch must be minimal because it may affect the consistency of point on wave control .
Current Limiting Reactors
Purpose: located between the generator and transformer (step up the voltage from voltage source to the secondary side. Not necessary its usage if the voltage source already had desired performance) together with resistors, so that before the current enter the test circuit, to control the current within limits together with the circuit power factor 
Test Breaker (TB)
Purpose: the tested circuit breaker. It might be a single phase or multi-phase depend on synthetic test circuit used.
TRV shaping circuit
Purpose: to control the transient recovery voltage (TRV) and rate of raise restriking voltage (RRRV).
Charged Capacitor Bank (High Voltage Source)
Purpose: to supply high voltage to the test circuit breaker. Banks of capacitors of up to hundreds of MVAR are required for the testing of CBs for the switching of shunt capacitor installations and are also used to test for overhead line and cable no-load switching. Capacitors are also required for the control of the circuit natural frequency for normal short circuit tests . The capacitor bank is modeled so that it could supply 800 kV.
Triggered Spark Gap
Purpose: to apply high voltage to the TB at desired moment. It controls the instant of injection in both current and voltage schemes.
Artificial Short Line (SL)
Purpose: to simulate the distance between the fault located on an overhead line from a switching station i.e. to simulate the varied conditions presented by single and bundle conductors. The majority of testing stations use an artificial line comprising various series-parallel combinations of inductance and capacitance.
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