Demand response (DR) has been recognized as a powerful tool to relieve energy imbalance in the smart grid. Most previous works have ignored the irrational behavior of energy consumers in DR project implementation. Accordingly, in this paper, we focus on solving two questions during the execution of DR. Firstly, considering the bounded rationality of residential users, a population dynamic model is proposed to describe the decision behavior on whether to participate in the DR project, and then the evolutionary process of consumers participating in DR is analyzed. Secondly, for the DR participants, they have to compete dispatching amounts for maximal profit in a day-ahead bidding market, hence, a non-cooperative game model is proposed to describe the competition behavior, and the uniqueness of the Nash equilibrium is analyzed with mathematical proof. Then, the distributed algorithm is designed to search the evolutionary result and the Nash equilibrium. Finally, a case study is performed to show the effectiveness of the formulated models.

Let X, Y be two real normed spaces with dimension $X\geq 2$ and $f:X\rightarrow Y$ ( $f\neq 0$ ) be a map which preserves equality of phase-distance with $f(0)=0$ , i.e., $$ \{ \|f(x)+f(y)\|, \|f(x)-f(y)\| \}=\{ \rho (\|x+y\|), \rho (\|x-y\|) \},\; \forall \; x,y\in X, $$ for some function $\rho :[0,\infty )\rightarrow [0,\infty )$ . In this paper, we prove that if f is surjective or Y is strictly convex, then f is phase equivalent to a linear map. More precisely, $$ f=\lambda \sigma \cdot T, $$ where λ is a non-zero real number, $\sigma : X\rightarrow \{-1, 1\}$ is a phase function and $T: X\rightarrow Y$ is a linear isometry. Counter example for the case when dimension $X=1$ is also presented.