Real-time simulation facts about mathematical solvers (fixed-step vs variable-step) and ARTEMIS

Here are a list of facts helping to understand the main differences between variable-step and fixed-step solvers and when they should be used.

1) Fixed-step solvers does not solve algebraic loops. Variable-step solvers must be used for problems with algebraic loops.

2) Variable-step solvers can be faster than fixed-step solvers for continuous system with few disturbances since the solver can automatically increase the time-step when there are no disturbances and decreases the time-step when switching occur.

3) The speed advantage of variable-step solvers decreases when the system has a large number of power electronic switches with fast switching frequency. In many cases, fixed-step solvers will be 10 or 100 time faster than variable-step solvers. See the ref paper of Nicola Lechevin attached.

4) The main advantages of variable-step solvers is their accuracy, the capability to solve algebraic loops, numerical stability and the fact that users do not need to specify the time-step.

5) Users must however select the tolerance and the type of variable-step solvers for each type of problems, which may not be evident. Variable-step solvers are still a subject of advanced research.

6) The main advantages of variable-step solvers are obtained at the condition that the solver is allowed to use the processing time required to achieve the specified accuracy. The processing time cannot be easily controlled and varies for each time step.

7) Consequently, variable-step solvers cannot be used for hard real-time application requiring to maintain a fixed processing time.

8) Variable-step solvers can be used for some real-time applications if the user can control the number of iteration based on the elapsed processing time, but this is rather complex.

9) It is also very difficult or impossible to implement parallel processing with variable-step solvers.

10) Users who must do HIL real-time simulation must transform their models such that it will be stable with a fixed-step solver using a time-step larger than 7 to 10 us (the minimum time-step that we can achieve as of now with the fastest computers available) but ideally 25 to 50 us should be used for electromagnetic transients studies.

11) Typical fixed-step solvers are Euler (order 1), Tustin (order 2) and ARTEMIS (order 5). Tustin order-2 is the most popular for electrical circuit simulation because it is more stable than Euler and is easy to implement with the nodal technique.

12) Artemis order-5 solvers is usually more accurate and stable than order-2 Tustin solver. Tustin uses the first two terms of the Taylor series of the exponential approximation while ARTEMIS used the first five terms. See the attached technical paper. As far as we know, ARTEMIS order-5 solver is only used with eMEGAsim, which also support Tustin.

13) ARTEMIS is also optimized for parallel simulation so very large power systems can be simulated in parallel using the latest multi-core processors. Actual variable-step solvers that we know cannot be used for parallel simulation.

14) ARTEMIS is also optimized for real-time simulation by pre-computing the state-space matrices for each breaker status (up to 18 switches per subsystem).

15) ARTEMIS is also interfaced with OPAL-RT IGBT VSC power electronic converters enabling the simulation of power electronic systems with hundreds of switches in real-time using advanced interpolation techniques. Such technique enable to accelerate the simulaton by a very large factor as compared to variable-step solvers and to reach real-time.

In summary, both variable-step and fixed-step solvers have their limitation and advantages. Users must then select the best solvers for each applications. Fast and real-time parallel simulation of very complex power grids and power electronic systems requires the use of advanced fixed-step solvers. Variable-step solvers can be used for smaller systems for applications where the main criteria is the accuracy and not the execution time.

Some continuous systems with few disturbances can take advantage of variable-step solvers to accelerate the simulation. But, the simulation of very complex circuits with several power electronic switches creating switching events at every few micros will be very long to simulate with variable-step solvers. In such case, advanced parallel fixed-step solvers such as ARTEMIS coupled with optimized power electronic converters are required.

OPAL-RT is also developing new advanced solvers for power grids and distribution circuits with very short lines for phasor-mode parallel simulation.

TR10-30202-AR-1 eMEGAsim_vs_PSCAD_circuit_RLC_.pdf - Nicola Lechevin tech paper
Comparison of computing time with and without ARTEMIS.DOC - Artemis computation time ref tech paper