Algorithmic challenges in green data centers

With data centers being the supporting infrastructure for a wide range of IT services, their efficiency has become a big concern to operators, as well as to society, for both economic and environmental reasons. The goal of this thesis is to design energy-efficient algorithms that reduce energy cost while minimizing compromise to service. We focus on the algorithmic challenges at different levels of energy optimization across the data center stack. The algorithmic challenge at the device level is to improve the energy efficiency of a single computational device via techniques such as job scheduling and speed scaling. We analyze the common speed scaling algorithms in both the worst-case model and stochastic model to answer some fundamental issues in the design of speed scaling algorithms. The algorithmic challenge at the local data center level is to dynamically allocate resources (e.g., servers) and to dispatch the workload in a data center. We develop an online algorithm to make a data center more power-proportional by dynamically adapting the number of active servers. The algorithmic challenge at the global data center level is to dispatch the workload across multiple data centers, considering the geographical diversity of electricity price, availability of renewable energy, and network propagation delay. We propose algorithms to jointly optimize routing and provisioning in an online manner. Motivated by the above online decision problems, we move on to study a general class of online problem named "smoothed online convex optimization", which seeks to minimize the sum of a sequence of convex functions when "smooth" solutions are preferred. This model allows us to bridge different research communities and help us get a more fundamental understanding of general online decision problems.

Distributed optimization in power networks and general multi-agent systems

The dissertation studies the general area of complex networked systems that consist of interconnected and active heterogeneous components and usually operate in uncertain environments and with incomplete information. Problems associated with those systems are typically large-scale and computationally intractable, yet they are also very well-structured and have features that can be exploited by appropriate modeling and computational methods. The goal of this thesis is to develop foundational theories and tools to exploit those structures that can lead to computationally-efficient and distributed solutions, and apply them to improve systems operations and architecture.

Specifically, the thesis focuses on two concrete areas. The first one is to design distributed rules to manage distributed energy resources in the power network. The power network is undergoing a fundamental transformation. The future smart grid, especially on the distribution system, will be a large-scale network of distributed energy resources (DERs), each introducing random and rapid fluctuations in power supply, demand, voltage and frequency. These DERs provide a tremendous opportunity for sustainability, efficiency, and power reliability. However, there are daunting technical challenges in managing these DERs and optimizing their operation. The focus of this dissertation is to develop scalable, distributed, and real-time control and optimization to achieve system-wide efficiency, reliability, and robustness for the future power grid. In particular, we will present how to explore the power network structure to design efficient and distributed market and algorithms for the energy management. We will also show how to connect the algorithms with physical dynamics and existing control mechanisms for real-time control in power networks.

The second focus is to develop distributed optimization rules for general multi-agent engineering systems. A central goal in multiagent systems is to design local control laws for the individual agents to ensure that the emergent global behavior is desirable with respect to the given system level objective. Ideally, a system designer seeks to satisfy this goal while conditioning each agent’s control on the least amount of information possible. Our work focused on achieving this goal using the framework of game theory. In particular, we derived a systematic methodology for designing local agent objective functions that guarantees (i) an equivalence between the resulting game-theoretic equilibria and the system level design objective and (ii) that the resulting game possesses an inherent structure that can be exploited for distributed learning, e.g., potential games. The control design can then be completed by applying any distributed learning algorithm that guarantees convergence to the game-theoretic equilibrium. One main advantage of this game theoretic approach is that it provides a hierarchical decomposition between the decomposition of the systemic objective (game design) and the specific local decision rules (distributed learning algorithms). This decomposition provides the system designer with tremendous flexibility to meet the design objectives and constraints inherent in a broad class of multiagent systems. Furthermore, in many settings the resulting controllers will be inherently robust to a host of uncertainties including asynchronous clock rates, delays in information, and component failures.

Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Optimization and Control of Power Flow in Distribution Networks

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators.

Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory.

The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.

Distributed Control and Optimization for Communication and Power Systems

We are at the cusp of a historic transformation of both communication system and electricity system. This creates challenges as well as opportunities for the study of networked systems. Problems of these systems typically involve a huge number of end points that require intelligent coordination in a distributed manner. In this thesis, we develop models, theories, and scalable distributed optimization and control algorithms to overcome these challenges.

This thesis focuses on two specific areas: multi-path TCP (Transmission Control Protocol) and electricity distribution system operation and control. Multi-path TCP (MP-TCP) is a TCP extension that allows a single data stream to be split across multiple paths. MP-TCP has the potential to greatly improve reliability as well as efficiency of communication devices. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new algorithm Balia (balanced linked adaptation) which generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We have implemented Balia in the Linux kernel. We use our prototype to compare the new proposed algorithm Balia with existing MP-TCP algorithms.

Our second focus is on designing computationally efficient algorithms for electricity distribution system operation and control. First, we develop efficient algorithms for feeder reconfiguration in distribution networks. The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. We propose a heuristic algorithm that is based on the recently developed convex relaxation of the optimal power flow problem. The algorithm is efficient and can successfully computes an optimal configuration on all networks that we have tested. Moreover we prove that the algorithm solves the feeder reconfiguration problem optimally under certain conditions. We also propose a more efficient algorithm and it incurs a loss in optimality of less than 3% on the test networks.

Second, we develop efficient distributed algorithms that solve the optimal power flow (OPF) problem on distribution networks. The OPF problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. Traditionally OPF is solved in a centralized manner. With increasing penetration of volatile renewable energy resources in distribution systems, we need faster and distributed solutions for real-time feedback control. This is difficult because power flow equations are nonlinear and kirchhoff's law is global. We propose solutions for both balanced and unbalanced radial distribution networks. They exploit recent results that suggest solving for a globally optimal solution of OPF over a radial network through a second-order cone program (SOCP) or semi-definite program (SDP) relaxation. Our distributed algorithms are based on the alternating direction method of multiplier (ADMM), but unlike standard ADMM-based distributed OPF algorithms that require solving optimization subproblems using iterative methods, the proposed solutions exploit the problem structure that greatly reduce the computation time. Specifically, for balanced networks, our decomposition allows us to derive closed form solutions for these subproblems and it speeds up the convergence by 1000x times in simulations. For unbalanced networks, the subproblems reduce to either closed form solutions or eigenvalue problems whose size remains constant as the network scales up and computation time is reduced by 100x compared with iterative methods.

Search for Higgs Boson Production Beyond the Standard Model Using the Razor Kinematic Variables in pp Collisions at √s=8 TeV and Optimization of Higgs Boson Identification Using a Quantum Annealer

In the first part of this thesis we search for beyond the Standard Model physics through the search for anomalous production of the Higgs boson using the razor kinematic variables. We search for anomalous Higgs boson production using proton-proton collisions at center of mass energy √s=8 TeV collected by the Compact Muon Solenoid experiment at the Large Hadron Collider corresponding to an integrated luminosity of 19.8 fb^-1.

In the second part we present a novel method for using a quantum annealer to train a classifier to recognize events containing a Higgs boson decaying to two photons. We train that classifier using simulated proton-proton collisions at √s=8 TeV producing either a Standard Model Higgs boson decaying to two photons or a non-resonant Standard Model process that produces a two photon final state.

The production mechanisms of the Higgs boson are precisely predicted by the Standard Model based on its association with the mechanism of electroweak symmetry breaking. We measure the yield of Higgs bosons decaying to two photons in kinematic regions predicted to have very little contribution from a Standard Model Higgs boson and search for an excess of events, which would be evidence of either non-standard production or non-standard properties of the Higgs boson. We divide the events into disjoint categories based on kinematic properties and the presence of additional b-quarks produced in the collisions. In each of these disjoint categories, we use the razor kinematic variables to characterize events with topological configurations incompatible with typical configurations found from standard model production of the Higgs boson.

We observe an excess of events with di-photon invariant mass compatible with the Higgs boson mass and localized in a small region of the razor plane. We observe 5 events with a predicted background of 0.54 ± 0.28, which observation has a p-value of 10^-3 and a local significance of 3.35σ. This background prediction comes from 0.48 predicted non-resonant background events and 0.07 predicted SM higgs boson events. We proceed to investigate the properties of this excess, finding that it provides a very compelling peak in the di-photon invariant mass distribution and is physically separated in the razor plane from predicted background. Using another method of measuring the background and significance of the excess, we find a 2.5σ deviation from the Standard Model hypothesis over a broader range of the razor plane.

In the second part of the thesis we transform the problem of training a classifier to distinguish events with a Higgs boson decaying to two photons from events with other sources of photon pairs into the Hamiltonian of a spin system, the ground state of which is the best classifier. We then use a quantum annealer to find the ground state of this Hamiltonian and train the classifier. We find that we are able to do this successfully in less than 400 annealing runs for a problem of median difficulty at the largest problem size considered. The networks trained in this manner exhibit good classification performance, competitive with the more complicated machine learning techniques, and are highly resistant to overtraining. We also find that the nature of the training gives access to additional solutions that can be used to improve the classification performance by up to 1.2% in some regions.