Abstract. We investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration.Key words. turnpike property, optimal control, dissipativity, stabilizability, controllability, model predictive control AMS subject classifications. 49K30, 49K21, 93B051. Introduction. An optimal trajectory of a control problem is said to have the turnpike property if it first approaches an equilibrium state, stays close to it for a while and finally turns away from it again. The name turnpike property is motivated by the analogy of the behavior of the optimal trajectories to the strategy of driving from a point A to B on a road system consisting of highways ("turnpikes") and smaller roads. When the distance from A to B is sufficiently long, it is typically time optimal to first drive from A to the nearest highway (= move to the equilibrium), drive on the highway towards the nearest exit to B (= stay near the equilibrium) and then exit in order to reach B via smaller roads (= turn away from the equilibrium).The turnpike property has been studied at least since the work of von Neumann in 1945 [22] and Dorfman, Samuelson and Solow in 1958 [11, p. 331]. Since then it has been observed in many optimal control problems. There is a vast amount of literature on sufficient conditions for this phenomenon to hold, see, e.g., [9, Section 4.4] or [28], particularly in economics, see, e.g., [21] and the references therein. However, only very few references treat the case of exponential turnpike which we consider in this paper for nonlinear undiscounted discrete time optimal control problems without terminal constraints. Our main motivation for studying this property is its recently discovered importance for obtaining convergence results in economic model predictive control (MPC) without terminal constraints, see [13]. The particular interest in exponentially fast versions of the turnpike property is triggered by the fact that, compared to slower turnpike properties, the exponential turnpike property allows to conclude additional qualitative properties of the MPC closed loop solution, like trajectory convergence and approximate finite time optimal transient behavior, for details see Section 3, below. While some exponential turnpike theorems can be found in the literature, our approach extends these results in various ways, e.g., by assuming only strict
Abstract-In this paper, we present a hierarchical, iterative distributed optimization algorithm, and show that the algorithm converges to the solution of a particular global optimization problem. The motivation for the distributed optimization problem is the predictive control of a smart grid, in which the states of charge of a network of residential-scale batteries are optimally coordinated so as to minimize variability in the aggregated power supplied to/from the grid by the residential network. The distributed algorithm developed in this paper calls for communication between a central entity and an optimizing agent associated with each battery, but does not require communication between agents. The distributed algorithm is shown to achieve the performance of a large-scale centralized optimization algorithm, but with greatly reduced communication overhead and computational burden. A numerical case study using data from an Australian electricity distribution network is presented to demonstrate the performance of the distributed optimization algorithm.
Abstract:The recent rapid uptake of residential solar photovoltaic (PV) installations provides many challenges for electricity distribution networks designed for one-way power flow from the distribution company to the residential customer. In particular, for grid-connected installations, intermittent generation as well as large amounts of generation during low load periods can lead to a degradation of power quality and even outages due to overvoltage conditions. In this paper we present two approaches to mitigate these difficulties using small-scale distributed battery storage. The first is a decentralized model predictive control (MPC) approach while the second is a hierarchical distributed MPC approach using a so-called market maker. These approaches are validated and compared using data on load and generation profiles from customers in an Australian electricity distribution network.
We consider supplier development within a supply chain consisting of a single manufacturer and a single supplier. Because investments in supplier development are usually relationship-specific, safeguard mechanisms against the hazards of partner opportunism have to be installed. Here, formal contracts are considered as the primary measure to safeguard investments. However, formal contracts entail certain risks, e.g., a lack of flexibility, particular in an ambiguous environment. We propose a receding horizon control scheme to mitigate possible contractual drawbacks while significantly enhancing the supplier development process and, thus, to increase the overall supply chain profit. Our findings are validated by a numerical case study.
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