Angelos Georghiou

Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization

In recent years, decision rules have been established as the preferred solution method for addressing computationally demanding, multistage adaptive optimization problems. Despite their success, existing decision rules (a) are typically constrained by their a priori design and (b) do not incorporate in their modeling adaptive binary decisions. To address these problems, we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions, respectively. We then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed-integer optimization. We demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems. We provide global lower bounds and show that our approach is (i) practically tractable and (ii) provides high quality solutions that outperform alternative methods.

Robust optimal control with adjustable uncertainty sets

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. We demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.

A stochastic optimization approach to cooperative building energy management via an energy hub

Building energy management is an active field of research since the potential in energy savings can be substantial. Nevertheless, the opportunities for large savings within individual buildings can be limited by the flexibility of the installed climate control devices and the individual construction characteristics. The energy hub concept allows one to manage a collection of buildings in a cooperative manner, by providing opportunities for load shifting between buildings and the sharing of expensive but energy efficient equipment housed in the hub, such as heat pumps, boilers, batteries. Typically, control design for the buildings and the energy hub are done separately, underutilizing the potential flexibility provided by the interconnected system. To address these issues, we propose a unified framework for controlling the operation of the energy hub and the buildings it connects to. By modeling all exogenous disturbance parameters as stochastic processes, and by using state-space representation of the building dynamics, we formulate a multistage stochastic optimization problem to minimize the total energy consumption of the system in a cooperative manner. We solve the resulting infinite dimensional optimization problem using a decision rule approximation, and we benchmark its performance on a numerical study, comparing it with established solution techniques.

Balancing bike sharing systems through customer cooperation - a case study on London's Barclays Cycle Hire

A growing number of cities worldwide have been installing public bike sharing systems, offering citizens a flexible and “green” alternative of mobility. In most bike sharing systems, customers rent and return bikes at different stations, without prior notification of the system operator. As a consequence, bike systems often become unbalanced, leaving some stations either empty or full. In such a case, customers either cannot pick up or return their bikes, resulting in a low service level. Typically, system operators employ staff to manually relocate bikes using trucks, leading to considerable operational cost. In this paper, we describe various methods to balance bike sharing systems by actively engaging customers in the balancing process. In particular, we show that by appropriately sending “control signals” to customers requesting them to slightly change their intended journeys, bike sharing systems can be balanced without using staffed trucks. Through extensive simulations based on historical data from Londons Barclays Cycle Hire scheme, we show that simple control signals are sufficient to effectively balance the bike sharing system and offer service rates close to 100%.

The Power of Diversity: Data-Driven Robust Predictive Control for Energy-Efficient Buildings and Districts

The cooperative energy management of aggregated buildings has recently received a great deal of interest due to substantial potential energy savings. These gains are mainly obtained in two ways: 1) exploiting the load shifting capabilities of the cooperative buildings and 2) utilizing the expensive but energy-efficient equipment that is commonly shared by the building community (e.g., heat pumps, batteries, and photovoltaics). Several deterministic and stochastic control schemes that strive to realize these savings have been proposed in the literature. A common difficulty with all these methods is integrating knowledge about the disturbances affecting the system. In this context, the underlying disturbance distributions are often poorly characterized based on historical data. In this paper, we address this issue by exploiting the historical data to construct families of distributions, which contain these underlying distributions with high confidence. We then employ tools from data-driven robust optimization to formulate a multistage stochastic optimization problem, which can be approximated by a finite-dimensional linear program. We demonstrate its efficacy in a numerical study, in which it is shown to outperform, in terms of energy cost savings and constraint violations, established solution techniques from the literature. We conclude this paper by showing the significant energy gains that are obtained by cooperatively managing a collection of buildings with heterogeneous characteristics.

Convex approximation of chance-constrained MPC through piecewise affine policies using randomized and robust optimization

In this paper, we consider chance-constrained Stochastic Model Predictive Control problems for uncertain linear systems subject to additive disturbance. A popular method for solving the associated chance-constrained optimization problem is by means of randomization, in which the chance constraints are replaced by a finite number of sampled constraints, each corresponding to a disturbance realization. Earlier approaches in this direction lead to computationally expensive problems, whose solutions are typically very conservative both in terms of cost and violation probabilities. One way of overcoming this conservatism is to use piecewise affine (PWA) policies, which offer more flexibility than conventional open-loop and affine policies. Unfortunately, the straight-forward application of randomized methods towards PWA policies will lead to computationally demanding problems, that can only be solved for problems of small sizes. To address this issue, we propose an alternative method based on a combination of randomized and robust optimization. We show that the resulting approximation can greatly reduce conservatism of the solution while exhibiting favorable scaling properties with respect to the prediction horizon.

Flow-maximizing equilibria of the Cell Transmission Model

We consider the freeway ramp metering problem, based on the Cell Transmission Model and address the question of how well decentralized control strategies, e.g. local feedback controllers at every onramp, can maximize the traffic flow asymptotically under time-invariant boundary conditions. We extend previous results on the structure of steady-state solutions of the Cell Transmission Model and use them to optimize over the set of equilibria. By using duality arguments, we derive optimality conditions and show that closed-loop equilibria of certain decentralized feedback controllers, in particular the practically successful “ALINEA” method, are in fact globally optimal.

Alleviating tuning sensitivity in Approximate Dynamic Programming

The simplicity of Approximate Dynamic Programming offers benefits for large-scale systems compared to other synthesis and control methodologies. A common technique to approximate the Dynamic Program, is through the solution of the corresponding Linear Program. The major drawback of this approach is that the online performance is very sensitive to the choice of tuning parameters, in particular the state relevance weighting parameter. Our work aims at alleviating this sensitivity. To achieve this, we propose to find a set of approximate Q-functions, each for a different choice of the tuning parameters, and then to use the pointwise maximum of the set of Q-functions for the online policy. The pointwise maximum promises to be better than using only one of individual Q-functions for the online policy. We demonstrate that this approach immunizes against tuning errors through a stylized portfolio optimization problem.

Racing miniature cars: Enhancing performance using Stochastic MPC and disturbance feedback

We study and compare different Stochastic Model Predictive Control (MPC) approaches for driving miniature race cars, where the goal is to race the cars as fast as possible around a known track. Designing such controllers is generally challenging since the models are not entirely accurate due to linearization errors and model mismatch. Consequently, deterministic MPC tends to be optimistic, causing the cars to leave the race track, resulting in accidents. To mitigate these shortcomings, methods based on Stochastic MPC have been proposed that ensure constraint satisfaction with high probability. Furthermore, one can reduce conservatism of the solution by optimizing over feedback policies at the expense of increased computational time. While methods based on affine state feedback policies have been shown to perform well, their performance critically depends on the choice of the feedback matrix. One way to ensure computational tractability is to fix the feedback matrix in an a-priori computation which is typically obtained via trial-and-error. To overcome this issue, we investigate the benefits of disturbance feedback policies, which allows us to (indirectly) optimize over the state feedback matrices. We verify the benefits of disturbance feedback policies in simulations, and also implement a heuristic variant on the actual system in experiment. Both studies suggest that Stochastic MPC with disturbance feedback is an attractive alternative to existing methods, due to its ability to increase performance and robustness compared to deterministic methods.