2.4.1. Problem formulation

The NPV of the system, which has been explained in more detail in Section 2.3, is considered as the objective function: the variable xc on which the NPV depends is the set consisting of the nominal value of the power Pacc and energy Energacc of the ESS, as well as the position Pos where the ESS itself is installed along the track:

$$\mathbf{x}\_{\mathbb{C}} = \langle \mathbf{P}\_{\text{acc}}, \mathbf{Energy}\_{\text{acc}'}, \mathbf{Pos} \rangle \tag{20}$$

The study is focused on finding the solution that has the maximum NPV, so the optimization problem is formulated as follows:

$$Z = \max\_{\mathbf{x}\_{\mathbb{C}}} \text{NPV}(\mathbf{x}\_{\mathbb{C}}) \tag{21}$$

subject to the following constraints:

$$P\_{\text{acc}}^{\text{min}} \le P\_{\text{acc}} \le P\_{\text{acc}}^{\text{max}} \tag{22}$$

Energmin acc≤ Energacc≤ Energmax acc (23)

$$0 \le \text{Pos} \le \text{L}^{\text{max}} \tag{24}$$

$$\mathbf{V}\_{\rm LINE}^{\rm min} \le \mathbf{V}\_{\rm LINE} \le \mathbf{V}\_{\rm LINE}^{\rm max} \tag{25}$$

$$\mathbf{I}\_{\rm ESS}^{\rm min} \le \mathbf{I}\_{\rm ESS} \le \mathbf{I}\_{\rm ESS}^{\rm max} \tag{26}$$

$$\mathbf{V}\_{\rm ESS}^{\rm min} \le \mathbf{V}\_{\rm ESS} \le \mathbf{V}\_{\rm ESS}^{\rm max} \tag{27}$$

$$\text{SoC}^{\text{min}}\_{\text{ESS}} \le \text{SoC}\_{\text{ESS}} \le \text{SoC}^{\text{max}}\_{\text{ESS}}\tag{28}$$

where Lmax is the track length, VLINEmin and VLINEmax are the minimum and maximum allowable line voltage values, IESS, VESS and SoCESS are the current, voltage, and SoC of to the ESS, respectively calculated using the software described in Section 2.2 [37–39]; each of them is limited to its minimum and maximum value.

The optimization of the siting and sizing of the ESS is formulated as a non-linear integer problem (MINLP) whose non-linear objective function is evaluated by means of a simulation tool, which is able to verify the safe operating conditions of the d.c. feeder system, and it is not known in analytic form. Thus it fits in the class of black-box non convex MINLP problems which are among the most challenging optimization models. The solution of black box problems requires the use of derivative-free algorithms that do not require the derivatives not even in approximate form. Further non convexity of the functions makes it difficult to use exact method for integer programs.

The grea<sup>t</sup> computational difficulty and the computational time needed in evaluating the objective function sugges<sup>t</sup> using simple heuristic algorithms to explore the solution space quickly. Hence the optimization problem is addressed using a Particle Swarm Optimization-based algorithm as the solution method.
