2.1.2. Action Space

The action space *at* to fix voltage issues in the studied network consists in changing the active and reactive powers of DG units, i.e., <sup>Δ</sup>*Pg*,*<sup>t</sup>* and <sup>Δ</sup>*Qg*,*<sup>t</sup>* ∀*g* ∈ G, as well as adjusting the transformer tap ratio Δ*Tapt*.

$$a\_t = (\Delta P\_{1,t}, \dots, \Delta P\_{G,t}, \Delta Q\_{1,t}, \dots, \Delta Q\_{G,t}, \Delta Tap\_t) \tag{2}$$

The actual changes in active <sup>Δ</sup>*Pg*,*<sup>t</sup>* and reactive <sup>Δ</sup>*Qg*,*<sup>t</sup>* power levels initiated at time *t* (which will define the power output at time *t* + 1) are limited by the available power at time *t* + 1:

$$0 \le P\_{\mathcal{K}^t} + \Delta P\_{\mathcal{K}^t} \le \widetilde{P}\_{\mathcal{K}^{t+1}} \,\forall \mathcal{g} \in \mathcal{G} \tag{3}$$

$$
\underline{\Delta Q}\_{\mathfrak{g},t+1} \le \Delta Q\_{\mathfrak{J},t} \le \overline{\Delta Q}\_{\mathfrak{g},t+1} \,\forall \mathfrak{g} \in \mathcal{G} \tag{4}
$$

where *Pg*,*<sup>t</sup>* denotes the power level of (dispatchable) generator *g* at time *t*, while <sup>Δ</sup>*Qg*,*t*+<sup>1</sup> and <sup>Δ</sup>*Qg*,*t*+<sup>1</sup> determine the safe range of variation of reactive power of unit *g* around the operation point. Likewise, the variation Δ*Tapt* around the operation point *Tapt* of the transformer tap change is given by:

$$
\Delta Tap \le Tap\_t + \Delta Tap\_t \le \overline{\Delta Tap} \tag{5}
$$

where Δ*Tap* and Δ*Tap* are the physical limits of the on-load tap changer.

It should be noted that other types of control actions, such as changing the terminal voltage set-points of (medium-sized) conventional generators or switching shunt devices, could also be considered if such resources are available in the system.
