**5. Incorporation of the vehicle model into the optimization procedure, objectives and constraints**

The established Simulink® model previously explained in this chapter is integrated to the MATLAB-based ACO script to iteratively work in tandem for the optimization purpose. The framework considers set of defined control and component sizing constraints and objectives. The co-design optimization process includes two iterative phases linked into each other through an outer loop to consider finding optimized EMS and component sizes at the meantime as simplified in **Figure 9**.

Two objective functions corresponding to the fuel consumption and components cost are considered for control and sizing optimizations, respectively. For the control parameter optimization, the decision variable would be the previously introduced LPF denominator (τ). Therefore, for the EMS optimization the algorithm aims to search for the power sharing variable which minimizes the fuel consumption (FC) while satisfying the constraints.

$$\min \left( \text{FC} \right) = \min f\_1 = \min \int\_0^t \dot{m} dt \tag{17}$$

On the other hand, another objective function is used for the component sizing formed based on the cost of powertrain components considering their prices per

power unit. In other words, for the optimal sizing, the algorithm searches for the sizes which minimize the powertrain cost while satisfying the constraints. In this regard, the following formulation can be readily expressed for this objective function:

$$\min\left(\text{Cost}\_{power train}\right) = \min f = \underset{\text{size}}{\text{arg min}} \left(\text{\bf{e}C}\_{\text{ICE}} + \text{\bf{e}C}\_{\text{EM}} + \text{\bf{e}C}\_{\text{inv}} + \text{\bf{e}C}\_{\text{Batt}} + \text{\bf{e}C}\_{\text{conv}}\right) \tag{18}$$

where the cost, €, for each component is considered in Euros and can be calculated based on per-power unit price, *Qcomp*, of each component considering its size:

$$\text{εC}\_{comp} = \left(\text{Q}\_{comp}\right) \left(\text{size}\_{comp}\right) \tag{19}$$

The used per power unit price are given in **Table 3** for the ICE, the battery and the DC-DC converter while the inverter cost can be directly included based on the following equation.

$$\text{€C}\_{inv} = \text{13.26}(P)^{1.1718} \tag{20}$$

For minimization of the objective functions, the charge sustaining HEV is subjected to the following inequality constrains:

$$|\text{SoC}\_f - \text{SoC}\_i| < \varepsilon\_0 \tag{21}$$

$$\text{SoC}\_{\text{min}} - \varepsilon < \text{SoC}(t) < \text{SoC}\_{\text{max}} + \varepsilon \tag{22}$$

$$C\_-Rate(t) \ge -3; \text{Negative sign stands for charging}\tag{23}$$

where the sizes of components are bounded between the considered minimum and maximum values of the search space. Regarding the SoC, constraint in Eq. (21) indicates the charge sustaining requirement, and Eq. (22) stands for the allowable limits of the SoC over the total driving cycle. The constraint in Eq. (23) is considered based on LiFePO4 battery type chemistry to avoid sudden charges, to avoid fast aging of the battery pack, and to improve battery's lifetime and performance. It is notable that some constraints must be incorporated into the objective function as penalties to penalize the cost via adding (in minimization problems) or deducting (in maximization problems) a big enough penalty value when the constraint(s) is violated. This technique is useful to consider the inequality constraints which cannot be directly involved in the formulations of the objective function. As the optimization problem for both objectives are both minimization type here, the added penalty is considered.


**Table 3.**

*Per-power unit prices used in cost objective function.*

*Application of Ant Colony Optimization for Co-Design of Hybrid Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.97559*
