Steady-State Error Compensation for Reinforcement Learning-Based Control of Power Electronic Systems

Read the Paper Link will follow when published

Reference tracking problems are common in power grids as well as in automotive applications, and the utilization of RL controllers in such scenarios feature better performance than state-of-the-art methods during transients. However, they still exhibit non vanishing steady-state errors. Therefore, the presented Steady-state Error Compensation tool will extend an established actor-critic based RL approach to compensate the steady-state error and close this gap.

Suggestions or experiences concerning applications of SEC are welcome!

Application Examples - Models

Electrical Power Grid Application

The first exemplary use case to show the benefits of the SEC approach is an electrical power grid application. The OpenModelica Microgrid Gym toolbox is used to model the control plant. The same setting as in this paper is used.

The dynamics of the shown average model of an inverter and a standard LC filter configuration (L_f, C_f), with inductor filter current i_f, voltage over the filter capacitor v_c and output load current i_load can be linearly described as:

with the switch voltage v_i. Above, R_f is the inductors' internal resistance, while it is assumed that the internal resistance of the capacitors is negligible. Moreover, the model can be described linearly in the state space

where x contains the states, the inputs are u and the load current can be interpreted as a disturbance:

The indices abc refer to a the three phases of a common three-phase electrical power grid (Compare for the gray 2 other phases shown in the figure above).

The corresponding state-space matrices are

Above, C is the 6x6 identity matrix assuming that the filter currents and voltages are measured while the load currents are not available as a measurement signals.

So far, the relevant currents and voltages are represented as vectors in a fixed abc reference frame. Hence, in steady state they are rotating with the frequency of the sinusoidal supply voltage. Using the Park transformation, the system variables can be mapped into a rotating reference frame. Here, the d-axis is aligned with the a-axis of the rotating three-phase system, the q-axis is orthogonal to the d-axis and the third is the zero component:

If the angular speed omega of the rotating frame is set equal to the grid frequency, the balanced sinusoidal grid voltages and currents become stationary DC-variables. This simplifies the control design, allowing for the effective application of linear feedback controllers. More information on the basics of power electronic control we refer to standard textbooks. This transformation is used in the calculations to perform the control in the rotation dq reference frame.

As baseline for the investigations a PI controller is used. It is implemented as a cascade of voltage and current controllers with configuration

These parameters are calculated using the in this paper described safe Bayesian optimization. 100 samples are drawn resulting in a performance increase of about 43 %, compared to the analytical controller layout.

Electrical Drive Application

An electric drive application is used as the second example. The Gym Electric Motor toolbox is used to model the control plant.

Like shown in the figure the plant is modeled in the rotation dq reference frame, too (For more information see power grid example explanation).

Again, the model can be described linearly in the state space:

where again x contains the states and the inputs are u:

The corresponding state-space matrices are

Here, omega describes the angular speed of the motor, multiplied with the magnetic flux (psi) resulting in the electromagnetic force (EMF) and p the number of pol pairs. C is the 2x2 identity matrix assuming that the all states are measurable. Here, the disturbance E is dependent on the rotor speed - which is chosen to be constant in the application.

As baseline for the investigations again a PI controller is used to control the currents in the rotation dq reference frame. The parameters are determined with a design tool provided by GEM (based on the symmetrical optimum) and are chosen as follows:

Citing

Detailed informations can be found in the article "Steady-State Error Compensation in Reference Tracking Problems with Reinforcement Learning Control". Please cite it when using the provided code:

@misc{weber2022,
      title={Steady-State Error Compensation in Reference Tracking and Disturbance Rejection Problems for Reinforcement Learning-Based Control}, 
      author={Daniel Weber and Maximilian Schenke and Oliver Wallscheid},
      year={2022},
      eprint={2201.13331},
      archivePrefix={arXiv},
      primaryClass={eess.SY}
      }

Usage

The jupyter notebook show_results.ipynp provides executable files to reproduce the plots and results shown in the paper.

Use

run_OMG_experiment()   

run_GEM_experiment()

to visualize the results. Like shown in the notebook

optuna_optimize_sqlite(ddpg_objective, study_name=STUDY_NAME, sampler=TPE_sampler)

can be used to train DDPG agents. In the respective (OMG/GEM / util) config file can be adjusted weather a standard DDPG ( env_wrapper='no_I_term') or an SEC-DDPG ( env_wrapper='past') should be trained. The data is logged depending on the loglevel to the in meas_data_folderdefined folder. Be aware that the calculation can be time consuming.