Optimal Control of Plug-in Vehicles Fleets in Microgrids

Abstract

This chapter focuses on the optimal operation of plug-in vehicle fleets in a microgrid characterized by the presence of other distributed resources, such as distributed generation units. The possible services that vehicle aggregators can provide in the microgrid are discussed and the control actions to be performed in order to obtain such services while integrating the vehicle aggregator actions with those of the other distributed resources of the grid are outlined. The problems with optimal operation are formulated as single-objective and multi-objective optimization problems, specifying, in both cases, objective functions, equality and inequality constraints. The differences and criticisms of both approaches are extensively analyzed in the chapter, where the two approaches are also implemented and solved with reference to the practical cases.

Appendices

Appendix 1

10.1.1 Weighted Sum Approaches

The weighted sum approach consists of solving the following single-objective optimization problem [20]:

$$\hbox{min} \sum\limits_{i = 1}^{m} {w_{i} F_{i} ({\mathbf{X}},{\mathbf{C}})}$$

(10.27)

subject to constraints (10.2) and (10.3). In (10.27), w _i are positive weights, whose values reflect the relative importance of the objectives. Typically, the weights satisfy the conditions $\sum\nolimits_{i = 1}^{m} {w_{i} = 1}$.

Before running the optimization algorithm, the DM indicates the preferences by setting the weights in (10.27). Regarding the relationship between the preferences and weights, even if the DM has a good knowledge of the objective functions and a precise idea of his own preferences, the solution obtained by minimizing (10.27) may not necessarily reflect the preferences [25]. The value of a weight must be relative to the other weights and relative to its corresponding objective function as outlined in [20]. Thus, the objective functions have to be transformed so that they all have similar magnitudes. As such, in this chapter, the following transformation in (10.27) is applied:

$$F_{i} = \frac{{f_{i} }}{{f_{i}^{\hbox{max} } }},$$

(10.28)

with $f_{i}^{\hbox{max} }$ the maximum value of the ith objective function. In this transformation F _i should result in a non-dimensional objective function with an upper limit of one and a lower limit of zero. To have a fast computation of $f_{i}^{\hbox{max} }$, for each objective function, the maximum value is approximated on the basis of an engineering intuition and computed in particularly critical grid operative conditions.

Several methods can be applied for the choice of the weights in (10.27). The type of method depends on the kind of information available from the DM. As a general rule, when the ratio scale properties of the DM evaluation of the objectives are available, the ratio weights methods should be used. When only the ordinal properties of the DM judgments are known, the rank order methods could be considered.

Ratio weights methods can be more effective than rank order weights methods but require extra time or effort to assess the ratio weights. Rank order methods are typically adopted when DMs are not able to provide more than ordinal information about objective function importance or when the decision process depends on the choice of a group, rather than one DM, whose members are able to agree on the ranking of the objectives, but not on the precise weights. Moreover, when choices require tradeoffs between objectives, the uncertainty regarding the preferences is even stronger [20, 25].

In this chapter, only the rank order methods were considered as criteria for choosing weights in the weighted sum approaches; in particular, the methods adopted were the rank order centroid (ROC) and rank sum (RS) methods [25].

In the ROC method a uniform distribution of the weights can be assumed on the simplex of the rank-order weights. Given that $w_{\left( 1\right)} > w_{( 2)} > \cdots > w_{{({\text{m}})}}$, if (i) is the rank position of w _(i) and m is the number of objective functions, the expected values of the true weights are given by:

$$w_{(i)} = \frac{1}{m}\sum\limits_{k = i}^{m} \frac{1}{k} \quad i = 1\,, \ldots\, , m.$$

(10.29)

In the RS method, the objective functions are ranked in terms of their relative importance. Each objective is then ranked in the proportion of its position in the rank order:

$$w_{(i)} = \frac{2(m + 1 - i)}{m(m + 1)} \quad i = 1\,, \ldots\, , m.$$

(10.30)

where: (i) is the rank position of the objective function i, $0 < w_{(i)} < 1$ and $\sum\nolimits_{i} {w_{(i)} \le 1}$.

It should be noted that, both the two approaches analyzed in this chapter (ROC and RS methods) are sharable and provide quite accurate results. The choice between the methods primarily depends on the particular circumstances. In more detail, as evidenced in [25], the RS weights are flatter than the ROC weights, then, the choice between these two approaches could depend also on the steepness of the true weights from which the DM’s decisions derive. The greater the value of the first few weights, the more attractive the ROC method.

It has to be observed that when dealing with a larger number of objectives, the differences among the methods are more pronounced.

10.1.2 Objective Sum Method

The objective sum (OS) method has no articulation of preferences. It can be stated that OS is a particular case of the weighted sum method with all weights being equal. In this way, it does not require any DM preference input. This is not necessarily a limit, because, in some cases, the DM has incomplete information and does not have a clear idea about their desires. At worst, the DM may have no preferences at all. The primary advantages of this method are its easy application and the reduced subjective requirements.

Appendix 2

The line parameters are reported in Table 10.4. The nominal values of active and reactive power requested by the loads are reported in Table 10.5.

Table 10.4 Line parameters

Table 10.5 Load active and reactive powers