Decentralized Charging Control for Large Populations of Plug-in Electric Vehicles: Application of the Nash Certainty Equivalence Principle Zhongjing Ma Duncan Callaway Ian Hiskens Abstract— The paper develops and illustrates a novel de- centralized charging control algorithm for large populations of plug-in electric vehicles (PEVs). The proposed algorithm is an application of the so-called Nash certainty equivalence principle (or mean-field games.) The control scheme seeks to achieve social optimality by establishing a PEV charging schedule that fills the overnight demand valley. The paper discusses implementation issues and computational complexity, and illustrates concepts with various numerical examples. Keywords: Decentralized control; plug-in electric vehicles (PEVs); Nash certainty equivalence (NCE) principle; ‘valley- fill’ charging control. I. I NTRODUCTION In order to reduce the emission of green-house gases and reliance on exhaustible petroleum sources, high penetrations of plug-in electric vehicles are expected to substitute the current conventional petroleum-combustion vehicles over the next few decades. The electricity demand from this large populations of PEVs may have a significant impact on the electrical power grid. For example, suppose that 30% of the 234 million conventional vehicles in the US were substituted by PEVs, that the average size of the PEV batteries is about 10 kWh, and that the charging rate of each PEV is about 2 kW. The total charging load would be 140 GW, which is 18% of the US summer peak load of 780 GW. Quite a few studies have been undertaken recently to explore the potential impacts of high penetrations of PEVs on the power grid [1], [2], [3], [4]. In [5], we study centralized optimal charging control, for large populations of homoge- neous PEVs. This work shows that under certain reasonable conditions, the control strategy results in valley filling, i.e. the total demand, consisting of aggregated PEV charging load and non-PEV demand, is constant during charging intervals. Note that all these proposals assume that the utility can directly control the charging rates of individual PEVs. The implementation of centralized charging control for large populations of PEVs is computationally intractable in general. It may also be impractical, due to the possible reluctance of PEV owners to allow their utility to directly control vehicle charging rates. In this paper, we suppose that Zhongjing Ma is with the Center for Sustainable Systems (CSS) and the School of Natural Resources and Environment, University of Michigan, e-mail: mazhong@umich.edu. Duncan Callaway is with the Energy and Resources Group, University of California, Berkeley, e-mail: dcal@berkeley.edu. Ian Hiskens is with the Department of Electrical Engineering and Com- puter Science, University of Michigan, email: hiskens@umich.edu. Research supported by the Michigan Public Service Commission through grant PSC-08-20, and the National Science Foundation through EFRI- RESIN grant 0835995. each of the PEV agents implements local charging controls for its own PEV. The (electricity) charging price, seen by all PEVs, is responsive to the total demand of the grid, which is the summation of the inelastic non-PEV base demand together with the aggregated charging rates of the whole population of PEVs. Because of the coupling through this common price signal, each PEV agent effectively interacts with the average charging strategy of the rest of the PEV population. As the population grows substantially, the influence of each individual PEV on that average charging strategy becomes negligible. Accordingly, for large populations, the average charging strategy seen by every PEV is identical. As a consequence, considering the charging controls for an infinite PEV population, a collection of local charging controls is a Nash equilibrium (NE), if (i) Each of the local controls is optimal with respect to one commonly observed charging trajectory, and (ii) The average of these local optimal charging controls is equal to the common trajectory, i.e. the average charging strategy is collectively reproduced by the local optimal control laws. This result is referred to as the Nash certainty equivalence (NCE) principle, as proposed by Huang et al. [6], [7]. This framework has connections with mean-field game models that were studied by Lasry and Lions [8], [9], and close con- nections with the notion of oblivious equilibrium proposed by Weintraub, Benkard, and Van Roy [10] via a mean-field approximation. Under certain reasonable conditions, the paper demon- strates via illustrative examples that there exists a Nash equilibrium. Moreover assuming that the electricity price is strictly increasing with respect to the total demand, it is verified that at a Nash equilibrium, the aggregated charging rates (of the PEV population) almost achieve ‘valley-fill’ (hence are nearly socially optimal.) For homogeneous PEVs, the charging control turns out to be a ‘valley-fill’ strategy. The paper is organized as follows. A class of PEV decentralized charging control problems is formulated in Section II. Section III introduces the so-called Nash certainty equivalence (NCE) principle. Using the NCE methodology, an algorithm is designed to implement the underlying de- centralized control strategy. The illustrative examples of Section IV demonstrate algorithm behaviour. The cost per- formance of the decentralized charging strategy is compared for different system specifications. Section V proposes an approach to handling unpredicted changes in system condi- tions. Conclusions are presented in Section VI, along with a 2010 IEEE International Conference on Control Applications Part of 2010 IEEE Multi-Conference on Systems and Control Yokohama, Japan, September 8-10, 2010 978-1-4244-5363-4/10/$26.00 ©2010 IEEE 191