A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
Abstract
:1. Introduction
- We study the demand response of the TCLs to control their set-point temperatures by considering the tradeoff between the electricity payment and TCL user’s comfort preference;
- According to the dynamics of TCLs, we set up the relationship between the energy demand and set-point temperature. Besides, we formulate the dissatisfaction function to represent the discomfort level of the set-point temperature;
- Based upon the interaction between EECC and TCL users, we formulate the specific energy trading process as a one-leader N-follower Stackelberg game;
- We show the existence and uniqueness of the equilibrium for the underlying Stackelberg games, and develop a DR algorithm based on the Backward Induction method to achieve the equilibrium.
2. Problem Formulation
2.1. TCL Dynamics
- and represent respectively the internal temperature () and the ambient temperature () of TCL i at time t.
- , and are thermal parameters which express the thermal resistance (kWh), thermal capacitance (/kW) and cooling thermal power (kW) of TCL i, respectively. For notational simplicity, we denote the thermal constant by , such that .
- The binary variable represents the switch state of TCL i at instant t.
2.2. Energy Demand of TCLs
- Case 1 (): By (1), we have the internal temperature at time , such that,
2.3. Energy Trading Process
3. Stackelberg Game Coordination
3.1. Stackelberg Game
- Stage I: Each TCL user i implements the best response function with respect to the broadcasted price p from EECC.
- Stage II: EECC optimizes the broadcasted price considering TCL users’ best response at Stage I.
- Leader level:
- Follower level:
3.2. Existence and Uniqueness of Stackelberg Equilibrium
Algorithm 1 DR algorithm by Backward Induction. |
Require:
Ensure:
|
4. Simulation
4.1. Homogeneous Case
4.2. Heterogeneous Case
5. Conclusions and Ongoing Reasearch
Author Contributions
Funding
Conflicts of Interest
Abbreviations
TCL | Thermostatically controlled loads |
EECC | Electric energy control center |
DR | Demand response |
RTP | Real-time pricing |
Appendix A. Proof of Lemma 2
References
- Siano, P. Demand response and smart grids—A survey. Renew. Sustain. Energy Rev. 2014, 30, 461–478. [Google Scholar] [CrossRef]
- Meng, F.L.; Zeng, X.J. An optimal real-time pricing for demand-side management: A Stackelberg game and genetic algorithm approach. In Proceedings of the 2014 International Joint Conference on Neural Networks, Beijing, China, 6–11 July 2014; pp. 1703–1710. [Google Scholar]
- Ipakchi, A.; Albuyeh, F. Grid of the future. IEEE Power Energy Mag. 2009, 7, 52–62. [Google Scholar] [CrossRef]
- Molderink, A.; Bakker, V.; Bosman, M.G.C.; Hurink, J.L.; Smit, G.J.M. Management and Control of Domestic Smart Grid Technology. IEEE Trans. Smart Grid 2010, 1, 109–119. [Google Scholar] [CrossRef] [Green Version]
- Callaway, D.S. Tapping the energy storage potential in electric loads to deliver load following and regulation, with application to wind energy. Energy Convers. Manag. 2009, 50, 1389–1400. [Google Scholar] [CrossRef]
- He, H.; Sanandaji, B.M.; Poolla, K.; Vincent, T.L. Aggregate Flexibility of Thermostatically Controlled Loads. IEEE Trans. Power Syst. 2013, 30, 189–198. [Google Scholar]
- Borenstein, S. The Long-Run Efficiency of Real-Time Electricity Pricing. Energy J. 2005, 26, 93–116. [Google Scholar] [CrossRef]
- Maharjan, S.; Zhu, Q.; Zhang, Y.; Gjessing, S.; Başar, T. Demand Response Management in the Smart Grid in a Large Population Regime. IEEE Trans. Smart Grid 2015, 7, 189–199. [Google Scholar] [CrossRef]
- Yu, M.; Hong, S.H. Supply-demand balancing for power management in smart grid: A Stackelberg game approach. Appl. Energy 2016, 164, 702–710. [Google Scholar] [CrossRef]
- Ma, Z.; Zou, S.; Ran, L.; Shi, X.; Hiskens, I.A. Efficient decentralized coordination of large-scale plug-in electric vehicle charging. Automatica 2016, 69, 35–47. [Google Scholar] [CrossRef]
- Dai, Y.; Gao, Y.; Gao, H.; Zhu, H. Real-time pricing scheme based on Stackelberg game in smart grid with multiple power retailers. Neurocomputing 2017, 260, 149–156. [Google Scholar] [CrossRef]
- Mortensen, R.E.; Haggerty, K.P. A stochastic computer model for heating and cooling loads. IEEE Trans. Power Syst. 1988, 3, 1213–1219. [Google Scholar] [CrossRef]
- Ucak, C.; Caglar, R. The effects of load parameter dispersion and direct load control actions on aggregated load. In Proceedings of the 1998 International Conference on Power System Technology, Beijing, China, 18–21 August 1998; Volume 1, pp. 280–284. [Google Scholar]
- Bashash, S.; Fathy, H.K. Modeling and Control of Aggregate Air Conditioning Loads for Robust Renewable Power Management. IEEE Trans. Control Syst. Technol. 2013, 21, 1318–1327. [Google Scholar] [CrossRef]
- Koch, S.; Mathieu, J.L.; Callaway, D.S. Modeling and control of aggregated heterogeneous thermostatically controlled loads for ancillary services. In Proceedings of the Power Systems Computation Conference, Stockholm, Sweden, 22–26 August 2011. [Google Scholar]
- Ghanavati, M.; Chakravarthy, A. Demand-Side Energy Management by Use of a Design-Then-Approximate Controller for Aggregated Thermostatic Loads. IEEE Trans. Control Syst. Technol. 2017, 26, 1439–1448. [Google Scholar] [CrossRef]
- Mathieu, J.L.; Koch, S.; Callaway, D.S. State Estimation and Control of Electric Loads to Manage Real-Time Energy Imbalance. IEEE Trans. Power Syst. 2013, 28, 430–440. [Google Scholar] [CrossRef]
- Yu, M.; Hong, S.H. A Real-Time Demand-Response Algorithm for Smart Grids: A Stackelberg Game Approach. IEEE Trans. Smart Grid 2017, 7, 879–888. [Google Scholar] [CrossRef]
- Yang, P.; Tang, G.; Nehorai, A. A game-theoretic approach for optimal time-of-use electricity pricing. IEEE Trans. Power Syst. 2013, 28, 884–892. [Google Scholar] [CrossRef]
- Samadi, P.; Mohsenian-Rad, A.H.; Schober, R.; Wong, V.W.S.; Jatskevich, J. Optimal Real-Time Pricing Algorithm Based on Utility Maximization for Smart Grid. In Proceedings of the IEEE International Conference on Smart Grid Communications, Gaithersburg, MD, USA, 4–6 October 2010; pp. 415–420. [Google Scholar]
- Tushar, W.; Chai, B.; Yuen, C.; Smith, D.B. Three-Party Energy Management With Distributed Energy Resources in Smart Grid. IEEE Trans. Ind. Electron. 2015, 62, 2487–2498. [Google Scholar] [CrossRef] [Green Version]
- Osborne, M.J.; Rubinstein, A. A Course in Game Theory; MIT Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Ladurantaye, D.D.; Gendreau, M.; Potvin, J.Y. Strategic Bidding for Price-Taker Hydroelectricity Producers. IEEE Trans. Power Syst. 2007, 22, 2187–2203. [Google Scholar] [CrossRef]
- Conejo, A.J.; Nogales, F.J.; Arroyo, J.M. Price-Taker Bidding Strategy under Price Uncertainty. IEEE Power Eng. Rev. 2002, 22, 57. [Google Scholar] [CrossRef]
- Liu, M.; Shi, Y. Model Predictive Control for Thermostatically Controlled Appliances Providing Balancing Service. IEEE Trans. Control Syst. Technol. 2016, 24, 2082–2093. [Google Scholar] [CrossRef]
- Barata, F.A.; Igreja, J.M.; Rui, N.S. Demand Side Management Energy Management System for Distributed Networks; Springer International Publishing: Basel, Switzerland, 2016; pp. 455–471. [Google Scholar]
- Perfumo, C.; Braslavsky, J.H.; Ward, J.K. Model-Based Estimation of Energy Savings in Load Control Events for Thermostatically Controlled Loads. IEEE Trans. Smart Grid 2014, 5, 1410–1420. [Google Scholar] [CrossRef]
- Yong, T.Y.; Jin, Y.G. Methods for Adding Demand Response Capability to a Thermostatically Controlled Load with an Existing On-off Controller. J. Electr. Eng. Technol. 2015, 10, 755–765. [Google Scholar] [Green Version]
- Tsui, K.M.; Chan, S.C. Demand Response Optimization for Smart Home Scheduling Under Real-Time Pricing. IEEE Trans. Smart Grid 2012, 3, 1812–1821. [Google Scholar] [CrossRef]
- Meng, F.L.; Zeng, X.J. A Stackelberg game-theoretic approach to optimal real-time pricing for the smart grid. Soft Comput. 2013, 17, 2365–2380. [Google Scholar] [CrossRef]
- Maharjan, S.; Zhu, Q.; Zhang, Y.; Gjessing, S.; Basar, T. Dependable Demand Response Management in the Smart Grid: A Stackelberg Game Approach. IEEE Trans. Smart Grid 2013, 4, 120–132. [Google Scholar] [CrossRef]
- Mathieu, J.L.; Callaway, D.S. State Estimation and Control of Heterogeneous Thermostatically Controlled Loads for Load Following. In Proceedings of the Hawaii International Conference on System Science, Maui, HI, USA, 4–7 January 2012; pp. 2002–2011. [Google Scholar]
i | Index of the TCL, |
Time interval | |
p | Broadcast price from the EECC |
Set-point temperature of TCL user i | |
Energy demand of TCL i in | |
Value of RTP | |
Internal temperature of TCL user i | |
Ambient temperature of TCL user i | |
Thermal resistance of TCL i | |
Thermal capacitance of TCL i | |
Switch state of TCL i at instant t | |
Temperature deadband | |
T | Length of the time interval |
Length of the “on” state in | |
Maximum energy demand in | |
Reference temperature of TCL user i | |
Reference energy demand of TCL user i in | |
j | Case of the switch state , |
Priority factor of the TCL user i |
Parameter | Homogeneous TCL | Heterogeneous TCL |
---|---|---|
R | /kW | /kW |
C | 5 kWh/ | 6 kWh/ |
P | 11 kW | 14 kW |
2.75 kWh | 3.5 kWh | |
b | 1.1 | 1.5 |
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Wang, P.; Zou, S.; Wang, X.; Ma, Z. A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads. Appl. Sci. 2018, 8, 1370. https://doi.org/10.3390/app8081370
Wang P, Zou S, Wang X, Ma Z. A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads. Applied Sciences. 2018; 8(8):1370. https://doi.org/10.3390/app8081370
Chicago/Turabian StyleWang, Peng, Suli Zou, Xiaojuan Wang, and Zhongjing Ma. 2018. "A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads" Applied Sciences 8, no. 8: 1370. https://doi.org/10.3390/app8081370