*2.3. Utility Model of Energy Consumption*

A residential community receives utility when it consumes energy in its own ways. When the energy consumption of the community is scheduled by the dispatching center of the smart grid, consumption utility will be affected. In order to quantitatively measure the utility, a utility model needs to be formulated. In many DR studies [27,28], quadratic and logarithmic utility functions are

frequently used, because they are non-decreasing and their marginal benefits are non-decreasing. In this paper, without loss of generality, the quadratic function is adopted as the utility model. That is:

$$u\_n^{h,t} = c\_h^t \left( L\_n^{h,t} \right)^2 + d\_h^t L\_n^{h,t} \tag{6}$$

where *c<sup>t</sup> <sup>h</sup>* <sup>&</sup>gt; 0 and *dt <sup>h</sup>* > 0 are time-varying parameters. Utility Equation (6) shows that, when a residential community shaves *Lh*,*<sup>t</sup> <sup>n</sup>* energy, then the utility will lose *uh*,*<sup>t</sup> <sup>n</sup>* . Therefore, the whole utility of community *n* in all time slots *T* can be calculated as:

$$u\_n^h = \sum\_{t=1}^T u\_n^{h,t} \tag{7}$$

Utility Equation (7) shows that community *n* will lose utility *u<sup>h</sup> <sup>n</sup>* when it shaves *<sup>T</sup> t*=1 *Lh*,*<sup>t</sup> <sup>n</sup>* energy in the daily dispatching period.
