# Ornstein–Uhlenbeck process

Auxiliary | Description | |
---|---|---|

1 | "Ornstein–Uhlenbeck" | Randomise the electrical network's dynamics |

## Summary

- In practicality an energy network is always in flux.
- The stochastic nature of the the loads and renewable generation can be simulated with ElectricGrid.jl.
- This notebooks will show you how to add stochastic reference signals via Ornstein–Uhlenbeck processes.

The Ornstein-Uhlenbeck process is described by the following stochastic differential equation (SDE):

\[ dX_t = κ(γ-X_t)dt + σdW_t\]

Here, $γ$ is the long term mean, $κ$ is the mean reversion parameter, $W_t$ denotes a Wiener process, and $σ$ scales the random impact.

`using ElectricGrid;`

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### Network Configuration

The following grid example is configured with

- Two inverters, a load, and two cables.
- The one inverter is placed in the familiar PQ mode, while the other is in VSG mode.
- The active and reactive power set points are confired as Ornstein–Uhlenbeck processes.
- The simulation is run for multiple episodes.

```
t_end = 0.2 # total run time, seconds
num_eps = 4 # number of episodes to run
# Connectivity Matrix
CM = [ 0. 0. 1.
0. 0. 2.
-1. -2. 0.]
# Nominal 3-phase power rating [VA], power factor, nominal rms voltage
R_load, L_load, _, _ = ParallelLoadImpedance(10e3, 0.95, 230)
parameters = Dict{Any, Any}(
"source" => Any[
Dict{Any, Any}("pwr" => 200e3,
"mode" => "VSG")
Dict{Any, Any}("pwr" => 100e3,
"mode" => "PQ",
"p_set" => -30e3, # Asymptotoic Mean
"q_set" => 10e3,
"std_asy" => 2.5e3, # Asymptotic Standard Deviation
"σ" => 100e3, # Random impact scale
"Δt" => 0.01, # Process time Step, seconds
"X₀" => 0.0, # Initial Process Value, Watt
"k" => 0) # Interpolation degree, {0, 1, 2}
],
"load" => Any[
Dict{Any, Any}("impedance" => "RL",
"R" => R_load,
"L" => L_load),
],
"cable" => Any[
Dict{Any, Any}("R" => 0.1,
"L" => 0.25e-3,
"C" => 0.1e-4),
Dict{Any, Any}("R" => 0.1,
"L" => 0.25e-3,
"C" => 0.1e-4),
],
);
env = ElectricGridEnv(CM = CM, parameters = parameters, t_end = t_end, verbosity = 2);
agents = SetupAgents(env);
hook = Simulate(agents, env, num_episodes = num_eps, );
```

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### Rendering

```
RenderHookResults(hook = hook,
episode = 2,
states_to_plot = [],
actions_to_plot = [],
power_p_inv = [2], # Real power [Watts]
power_q_inv = [2], # Imaginary power [VAi]
)
```

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### Analysis

- The active and reactive powers of the PQ load fluctuate around their mean set points.
- Increasing the interpolation degree to either 1 or 2 will remove the "steps" and smooth the output powers.
- The user can render all 4 episodes, which will show different dynamics.