Brief paperEvolutionary game theoretic demand-side management and control for a class of networked smart grid☆
Introduction
Demand-side management of energy systems becomes increasingly popular, because of its great potential in improving energy efficiency in industries. Smart grid is a typical platform where demand-side management strategies can be applied. A core issue in smart grid is that, dynamic user behaviors should be addressed in designing demand-side management strategies. Widely-used techniques for demand-side management of smart grid include game theoretic approach (Mohsenian-Rad, Wong, Jatskevich, Schober, & Leon-Garcia, 2014), multi-objective optimization (Malatji et al., 2013, Nwulu and Xia, 2015), distributed energy consumption control (Ma, Hu, & Spanos, 2014), and model predictive control (Zhang & Xia, 2011), etc.
Smart grids can be analyzed in the perspective of network systems, since there usually exist multiple interactive users consuming powers from grids. In networked smart grid systems, stability and optimization are two main issues. Stability of the networked smart grid system indicates that interactive users reach an equilibrium. Some methodologies, i.e. game theory (Mohsenian-Rad et al., 2014), can be applied to prove the existence of equilibria in networked smart grid system. Optimization of the network smart grid system implies that, in the transient process to reach the equilibrium, some indexes can be optimized. The grid provider is capable of influencing decisions of users in the network by presenting dynamic pricing strategies (Jiang et al., 2014, Li et al., 2013). It is possible that the smart grid provider and some of the users cooperate to affect decisions of other users, such that the common benefit can be improved.
Game theory has been widely applied to energy systems (Du et al., 2015, Hong et al., 2014). In previous researches on game theoretic policy for energy systems, fundamental games are usually played between two individual users (Xiao, Mandayam, & Poor, 2015), or between the power company and users (Fadlullah, Quan, Keto, & Stojmenovic, 2014). Pay-off functions and strategies are usually defined such that existence of Nash Equilibrium (NE) can be proved. Optimization (or model predictive control Stephens, Smith, & Mahanti, 2015) can be employed to search for NE. Sometimes the fundamental game is played repeatedly, and strategies of users are updated in real-time. In this situation, it is named evolutionary game (Cheng, He, Qi, & Xu, 2015). Networked evolutionary game indicates that, the repeated game is played among networked users, and updating laws relate to topological structure of the network (Cheng, 2009). In some networked evolutionary games, actions of some users can be actively assigned, such that other users are induced to improve common benefit. The users with actively assigned actions can be defined as controllers; and the networked evolutionary game with controllers can be defined as control networked evolutionary game (Zhao, Li, & Cheng, 2011).
During recent years, a new semi-tensor product (Cheng, Qi, & Xue, 2007) is developed to solve the problem of networked evolutionary game. The semi-tensor product is an extension of ordinary matrix product. By using semi-tensor product, dynamics of evolutionary games can be formulated into an algebraic form (Cheng, 2009), and the existence of NE can be proved systematically (Cheng et al., 2015). For the control networked evolutionary game, control strategies can be designed to reach the NE by using semi-tensor product. Moreover, classical control methods can be introduced and extended in the framework of semi-tensor product to attain the NE of the networked evolutionary game.
In this paper, demand-side management of a class of smart grid is studied within the framework of control networked evolutionary game. The smart grid is built among interactive communities using either grid power or local generated power. It is assumed that a small portion of the communities are subsidized, thus cooperative with the grid provider. However, other communities are un-subsidized and pursuing individual benefits. We aim to design actions for the cooperative communities (controllers), such that the common benefits can be improved even if other communities are noncooperative. The main contributions of this paper include that: (1) the demand-side management of a smart grid is modeled into a control networked evolutionary game; (2) the networked evolutionary game is composed by fundamental games played simultaneously among several players instead of 2-player games; (3) semi-tensor product is applied to solve the demand-side management problem; and (4) a new binary optimal control is introduced to optimize the transient performance of the control networked evolutionary game.
The layout of this paper is arranged as follows. In Section 2, mathematical preliminaries are introduced. In Section 3, the demand-side management of a simple smart grid is formulated within the framework of control networked evolutionary game. In Section 4, the proposed control evolutionary game is analyzed and solved by using semi-tensor product, and a new optimal control approach is proposed to improve transient performance. In Section 5, a simulation example is presented to illustrate the proposed demand-side management approach. This paper is concluded in the final section.
Section snippets
Control networked evolutionary game
Information interchange within networked system can be described by a directed graph , where is a set of nodes, and is a set of edges that depict information flow between nodes. An edge in denotes that the information of node is available to , and is defined as a neighbor of . The index set of all neighbors of node is denoted by . In an undirected graph, . The adjacent matrix , where if
Problem formulation
In this paper, the evolutionary game is played among some remote rural communities, where a networked power grid is newly constructed. Before the construction of the power grid, the communities were using power generated by local facilities, e.g., diesel generators. To cover the cost, the price of grid power is high when there are less users. As the number of users grows, the price of grid power would decrease. However, if the number of users grows excessively large, the price would increase
Algorithm for calculating the algebraic form
Suppose that there are communities, and their communication topology can be given by adjacent matrix . The algorithm for calculating the algebraic form (3) can be designed as follows.
- i.
Set .
- ii.
Set initial value . Based on the initial value, calculate the grid power price and the price .
- iii.
Use and to calculate the cost function .
- iv.
Based on , and the updating law, the updated strategy can be obtained: .
- v.
Set , and go to ii until .
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Calculate
A simulation example
In this section, an illustrative example is presented by considering a smart grid with more communities. Its topological structure is given by an undirected circular network with 10 nodes, as can be seen from Fig. 2. Based on concepts in Section 2.1, its adjacent matrix can be calculated accordingly. The diesel power price is given by , and the grid power price is given by , where is the number of communities using the grid power. In this simulation example, and
Conclusion
In this paper, control networked evolutionary game and semi-tensor product are applied to solve the demand-side management problem of a simple smart grid. By using the semi-tensor product to solve the control networked evolutionary game, NEs can be proved systematically, and control series can be designed to reach and maintain the optimal NE. The BNFO algorithm is introduced to optimize the transient performance of the control networked evolutionary game.
Some future works of this research
Bing Zhu was born in Wuhan, China, in 1985. He received his B.S. and Ph.D. degrees in control theory and applications from Beihang University (formerly named Beijing University of Aeronautics and Astronautics), Beijing, China P.R., in 2007 and 2013, respectively. He has been with University of Pretoria, Pretoria, South Africa, as a postdoctoral fellow with Vice-Chancellor Postdoctoral Fellowship from 2013 to 2015. He is currently a research fellow at Nanyang Technological University, Singapore.
References (20)
- et al.
LPV modeling and game-theoretic control synthesis to design energy-motion regulators for electric scooters
Automatica
(2014) - et al.
Control parameterization enhancing transform for optimal control of the switched systems
Mathematical and Computer Modelling
(2006) - et al.
A multiple objective optimisation model for building energy efficiency investment decision
Energy and Buildings
(2013) - et al.
Multi-objective dynamic economic emission dispatch of electric power generation integrated with game theory based demand response programs
Energy Conversion and Management
(2015) - et al.
A model predictive control approach to the periodic implementation of the solutions of the optimal dynamic resource allocation problem
Automatica
(2011) Input-state approach to Boolean networks
IEEE Transactions on Neural Networks
(2009)- et al.
Modeling, analysis and control of networked evolutionary games
IEEE Transactions on Automatic Control
(2015) - et al.
A survey on semi-tensor product of matrices
Journal of Systems Science and Complexity
(2007) - et al.
Game-theoretic formulation of power dispatch with guaranteed convergence and prioritized best response
IEEE Transactions on Sustainable Energy
(2015) - et al.
GTES: an optimized game-theoretic demand side management scheme for smart grid
IEEE Systems Journal
(2014)
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Bing Zhu was born in Wuhan, China, in 1985. He received his B.S. and Ph.D. degrees in control theory and applications from Beihang University (formerly named Beijing University of Aeronautics and Astronautics), Beijing, China P.R., in 2007 and 2013, respectively. He has been with University of Pretoria, Pretoria, South Africa, as a postdoctoral fellow with Vice-Chancellor Postdoctoral Fellowship from 2013 to 2015. He is currently a research fellow at Nanyang Technological University, Singapore. His research interests include demand-side management, model predictive control, nonlinear control, flight control, and multi-agent systems.
Xiaohua Xia obtained his Ph.D. degree at Beijing University of Aeronautics and Astronautics, Beijing, China, in 1989. He stayed at the University of Stuttgart, Germany, as an Alexander von Humboldt fellow in May 1994 and for two years, followed by two short visits to Ecole Centrale de Nantes, France and National University of Singapore during 1996 and 1997, respectively, both as a post-doctoral fellow. He joined the University of Pretoria, South Africa, since 1998, and became a full professor in 2000. He also held a number of visiting positions, as an invited professor at IRCCYN, Nantes, France, in 2001, 2004 and 2005, as a guest professor at Huazhong University of Science and Technology, China, and as a Cheung Kong chair professor at Wuhan University, China. He is an IEEE fellow, served as the South African IEEE Section/Control Chapter Chair, as the chair of the Technical Committee of Non-linear Systems, as a member of the Technical Board (both of IFAC). He is an A-rated scientist by the National Research Foundation of South Africa, an elected fellow of the South African Academy of Engineering, and an elected member of the Academy of Science of South Africa. He has been an Associate Editor of Automatica, IEEE Transactions on Automatic Control, IEEE Transactions on Circuits and Systems II, and the Specialist Editor (Control) of the SAIEE Africa Research Journal. His research interests include: non-linear feedback control, observer design, time-delay systems, hybrid systems, modeling and control of HIV/AIDS, control and handling of heavy-haul trains and energy modeling and optimization. He is a registered professional engineering by the Engineering Council of South Africa, and a certified measurement and verification professional by the American Association of Energy Engineers. He is the director of both the Centre of New Energy Systems at the University of Pretoria and the National Hub for the Postgraduate Programme in Energy Efficiency and Demand Side Management. He is an elected board member of measurement and verification council of South Africa (MVCSA) since 2014. He is the founding director of Onga Energy Efficiency and Management Pty Ltd—the first SANAS accredited M&V Company against ISO 17020 and he is invited as a technical assessor for the South African National Accreditation Systems (SANAS) for M&V inspection bodies in South Africa.
Zhou Wu was born in Anhui, China, in 1985. He received his B.S and M.S. degrees in control engineering from Wuhan University, Wuhan, China P.R., in 2007 and 2009, respectively, and his Ph.D. degree in electronic engineering from City University of Hong Kong, Hong Kong, in 2012. Then he joined in University of Pretoria, Pretoria, South Africa, as a Senior Research Fellow in the Centre of New Energy Systems. Since 2016, he joined Chongqing University, Chongqing, as a full professor. His research interests include evolutionary algorithms, pattern recognition, energy system optimization, demand side management, and optimal control.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen.
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