Abstract: The smart grid has allowed the analysis of electricity user's consumption in real time. This development makes it possible to adopt the real-time pricing as a tool to encourage users to consume electricity more efficiently and wisely. From the social perspective, it is desired to increase the sum of comfort obtained by each user and to decrease the expense imposed to the supply side. This aim can be formulated as maximizing the sum of utility functions of all users minus the cost function of the supply side, while the energy demand is constrained by the supply capacity, which is said to be the social welfare maximization model. By computing the Lagrangian multiplier, i.e., shadow price, of the social welfare maximization model, the real-time price is obtained. Usually, the Lagrangian multiplier is computed by the duality method. In the duality method, the price and the demand interact with each other in a distributed manner, and finally converge to a win-win agreement,which is beneficial to both the supply side and all users. In the existing methods of the social welfare maximization of real-time pricing in smart grid, some models contain both the lower limit of power supply (i.e. the minimum power generation) and the upper limit of power supply constraints, and the other models only contain the upper limit of power supply constraint for the purpose of simplification.
Suppose that K denotes the time slots number, K is the number of users, x_i^k denotes power consumption for the customer i at time slot k, L_k denotes electricity power generated by providers, L_k^min and L_k^max are minimum and maximum power generated by provider respectively, and L_k denotes power generated by provider at time slot k,U_i(x_i^k,ω_i^k) denotes utility function for the user i at time slot k, C_k(L) is the cost of L units of energy at time slot k.The following is called the social welfare maximization problem:
$\begin{array}{l} \mathop {\max }\limits_{{L_k}^{\min } \le {L_k} \le {L_k}^{{\mathop{\rm m}\nolimits} {\rm{ax}}}} \;\sum\limits_{i = 1}^n {{U_i}} ({x_i}^k,{w_i}^k) - {C_k}({L_k})\\ {\rm{s}}.{\rm{t}}{\rm{.}}\;\;\;\sum\limits_{i = 1}^n {{x_i}^k \le } {L_k}\;\;\;\;\;\;\;\;\;\;\;\;\;(1) \end{array}$
In this paper, the role of the lower limit power supply constraint in the model of social welfare maximization is investigated. By introducing a so-called effective cost function $\{ {C_k}(\sum\limits_{i = 1}^N {{x_i}^k} ),{C_k}({L_k}^{\min })\} $, under a mild assumption, we proved that the problem (1) is equivalent to the following problem:
$\begin{array}{l} {\rm{max}}\sum\limits_{i = 1}^N {{U_i}} ({x_i}^k,{w_i}^k) - \max \{ {C_k}(\sum\limits_{i = 1}^N {{x_i}^k} ),{C_k}(L_k^{\min })\} \\ {\rm{s}}.{\rm{t}}{\rm{.}}\;\;\;\sum\limits_{i = 1}^N {{x_i} \le } L_k^{\max }\;\;\;\;\;\;\;\;(2) \end{array}$
In other words, the mode with both the lower limit of power supply and the upper limit of power supply constraints and the model with only the upper limit of power supply constraint are equivalent. The assumptionthat we used is reasonable in practical application. This means that the minimum power supply constraint in the social welfare maximization model can be removed. Thus,one interval constraint and one variable are reduced in the problem (2). Meanwhile, it is showed that the problem (2) still satisfies the requirements of the online dual optimization method. This study aims at the basic social welfare maximization model, and the conclusions obtained can be extended to various improved and extended social welfare maximization models.

Key words: smart gird, real-time pricing, social welfare maximization, shadow price, optimization