Mathematical Problems in Engineering

Volume 2019, Article ID 1498134, 18 pages

https://doi.org/10.1155/2019/1498134

## Distributed Real-Time Pricing Method Incorporating Load Uncertainty Based on Nonsmooth Equations for Smart Grid

School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China

Correspondence should be addressed to Yan Gao; nc.ude.tssu@nayoag

Received 21 September 2018; Accepted 19 December 2018; Published 12 February 2019

Academic Editor: Hong-Yu Wu

Copyright © 2019 Hongjie Wang and Yan Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The real-time pricing mechanism of smart grid based on demand response is an effective means to adjust the balance between energy supply and demand, whose implementation will impact the user's electricity consumption behaviour, the operation, and management in the future power systems. In this paper, we propose a complementarity algorithm to solve the real-time pricing of smart grid. The Karush–Kuhn–Tucker condition is considered in the social welfare maximisation model incorporating load uncertainty to transforming the model into a system of nonsmooth equations with Lagrangian multipliers, i.e., the shadow prices. The shadow price is used to determine the basic price of electricity. The system of nonsmooth equations is a complementarity problem, which enables us to study the existence and uniqueness of the equilibrium price and to design an online distributed algorithm to achieve the equilibrium between energy supply and demand. The proposed method is implemented in a simulation system composed of an energy provider and 100 users. Simulations results show that the proposed algorithm can motivate the users’ enthusiasm to participate in the demand side management and shift the peak loading. Furthermore, the proposed algorithm can improve the supply shortage. When compared with an online distributed algorithm based on the dual optimisation method, the proposed algorithm has a significantly lower running time and more accurate Lagrangian multipliers.

#### 1. Introduction

Given the increased expectations of customers, in both quality and quantity [1], the limited energy resources, and the lengthy and expensive process of exploiting new resources, the reliability of the grid has been put in danger and there is a need to develop new methods to increase the grid efficiency. Smart grid is an intelligent and efficiency power system that allows information exchange between users and power providers. Smart grid has demand response (DR) functionality, and provides a foundation to enable demand side activities in smart grid. The price-response based DR programs, especially, can effectively promote user enthusiasm to participate in the market, and it is one of the most cost-effective elements as regards energy cost reduction for residential and small industrial buildings [2, 3]. The DR programs primarily shift users’ consumption from peak to off-peak periods to reduce loads on utility-handling equipment, i.e., distribution transformers and lines, and may constitute a value resource for effective operation of the smart grid structure [4, 5]. Thus, the researches on pricing mechanism of smart grid based on demand response will impact the user's electricity consumption behaviour and the operation and management in the future power systems [6].

Currently, the main pricing methods based on DR programs contain fixed, multistep, time-of-use, and real-time pricing. The latter has advantages over the multistep and time-of-use pricing methods, which is the most effective for demand side management and can avoid new peaks caused by user electricity consumption during low-price periods [7–10]. When the real-time pricing method is implemented, users can shift their usage and save on electricity loads to reduce electricity costs while satisfying building inhabitant comfort level and satisfaction requirements. Notably, the participative behaviours of users are useful to achieve peak-load shifting. Bahrami et al. [10] have proposed a real-time scheduling problem of energy hubs in a dynamic pricing market. The simulations have validated that the dynamic pricing can increase the energy hubs’ average payoff. In 2010, Samadi et al. [11] have proposed a utility framework that is aimed at the total utility being maximised for all users and the production cost being minimised for the energy provider. The framework is converted into an optimisation problem containing a Lagrangian multiplier for users and provider in the dual domain. Samadi et al. used the Lagrangian multiplier to determine the basic price of electricity and designed a distributed algorithm for real-time pricing in smart grid. The studies on real-time pricing of smart grid mainly are in two aspects: improving the model and designing algorithm for the social welfare maximisation model, since 2010 [12–18]. Koji [12] has used Jensen’s inequality and the weighted arithmetic-geometric mean inequality to solve utility-maximisation problems. Zhu et al. [13] have proposed a distributed optimisation algorithm based on the alternating direction method of the multiplier algorithm with Gaussian back substitution, so as to obtain solutions of the real-time pricing model (i.e., the social welfare model). With consideration of electricity vehicle charging, Bitar [14] has proposed a novel forward market for a deadline-differentiated electric-power service-based utility function, in which consumers consent to defer service of prespecified loads in exchange for reduced energy prices. Song et al. [15] have proposed an improved algorithm based on the gradient projection method to solve the problem of poor convergence associated with optimal real-time pricing model based on utility maximisation. Asadi et al. [16] have proposed use of the particle swarm optimisation algorithm to solve the real-time pricing problem based on the social welfare model. Zhu et al. [17] have solved the real-time pricing problem using the simulated annealing algorithm. Furthermore, Tarasak [18] has extended the social welfare maximisation model to include the effect of the load uncertainty; this approach derives the optimal prices under different types of load uncertainty and demonstrates the influence of the load uncertainty on the power consumption and generating capacity. Bahrami et al. [19] have considered an electricity market with generation uncertainty and proposed a day ahead decentralised algorithm to solve the optimal energy trading.

The above techniques are based on dual optimisation. The dual optimisation method solves the price problem corresponding to the Lagrangian multipliers in the real-time pricing problem; however, the multiplier is obtained as a by-product, which affects the multiplier accuracy. In fact, the multiplier, i.e., price, is our concern, and price bias errors can cause various losses. Furthermore, the dual optimisation method is a minimax problem suitable for application to small-scale problems only. In the smart grid, the number of users is huge, and more power providers can be included in the future. Hence, the scale of the problem is large and satisfactory computation speed of real-time pricing cannot be guaranteed if the dual optimisation method is applied. In this study, we propose an algorithm for the real-time pricing of smart grid that can provide enhanced computation speed and accuracy over the dual optimisation method. We consider the Karush–Kuhn–Tucker (KKT) condition in the social welfare maximisation model and use a system of nonsmooth equations to solve the real-time pricing of smart grid. The contributions of this paper can be summarised as follows.

(1) Based on social welfare maximisation model, this paper establishes the real-time pricing model of smart grid by KKT condition. A system of nonsmooth equations with the Lagrangian multipliers is creatively established. The Lagrangian multipliers, i.e., the shadow prices, are used to determine the basic price of electricity. Compared to the dual method, the Lagrangian multiplier is a decision variable and plays the same role as electricity consumption and generating capacity level in the nonsmooth equations. Hence, the numerical results of Lagrangian multiplier are more accurate in the nonsmooth equations.

(2) The system of nonsmooth equations is converted into two optimal problems for users and provider. Then an online distributed algorithm is designed to solve the real-time pricing of smart grid. In the proposed algorithm, users can shift their usage and save electricity loads to reduce electricity costs while satisfying their requirements according to the prices provided by energy provider in each time slot, and the energy provider can adjust prices and electricity production levels according to users' demand and expectations. This approach is beneficial for shifting the load peak and avoiding a new load peak.

(3) The smooth Newton method is used to solve the nonsmooth equations, i.e., solving the real-time pricing in the proposed distributed algorithm. The Newton method is good convergence that ensures the computational speed of the algorithm. In the simulation experiments considering a power system with an energy provider and 100 users, the simulation results have proved that the algorithm is more accurate and the convergence is better than that of the dual method.

The remainder of this paper is organised as follows. In Section 2, we briefly describe the social welfare maximisation model, i.e., the real-time pricing model. In Section 3, a model with a KKT condition is established, instead of the social welfare model, and transformed into two subproblems. The smooth Newton method is selected to solve the equations. Finally, simulation results are presented and discussed in Section 4.

#### 2. Preliminary Knowledge

##### 2.1. Electricity System Model

A smart grid constitutes a delivery system composed of an energy provider and multiple users. The energy provider constantly contracts with and serves the users. In addition, each user installs a smart meter to control their energy consumption; hence, two-way real-time communication between the energy provider and user is allowed. The communication network delivers information required for smart grid operation, such as energy demand, electricity price, and customer identification data. The smart meter aggregates the power consumption demands of all appliances and reports the result to the energy provider. In return, the smart meter receives a real-time price from the energy provider and then user adjusts the appliance electricity consumption accordingly.

The notation employed in this study is the same as that used in [11]. A one-day operation cycle is divided into one-hour time slots. So, the real-time pricing and demand response have a one-hour resolution. Denote as the set of time slots, where an individual time slot . We use to represent the set of users requiring electricity, where a given user requires a certain amount of energy for his/her electricity appliances. We take as the electricity consumption demand of user at time slot . The electricity consumption for each time slot and user is bounded; i.e., . Here, the minimum electricity consumption level represents the load from the appliances that are required to be in constant operation throughout the day and night, while the maximum electricity consumption level represents the total power consumption level of the appliances, assuming they are all in operation simultaneously.

##### 2.2. Utility Function

Each user is an independent individual in the electricity system and has a unique satisfaction level. From the perspective of microeconomics, the utility function is a measure of a user’s demands and desires based on their consumption or leisure activities. We define the utility function of an individual as , where is the user’s electricity consumption. In previous studies, some researchers adopted a quadratic function as the utility function [9, 10]. In this work, we also use a quadratic function as the utility function, which is expressed asHere, is a predetermined parameter characterising the utility saturation point and is a nonnegative parameter characterising the user types assumed to be private. typically varies for different time slots and users. We can choose different by selecting the appropriate (see Figure 1).