Data privacy is emerging as one of the most serious concerns of big data analytics, particularly with the growing use of personal data and the ever-improving capability of data analysis. This dissertation first investigates the relation between different privacy notions, and then puts the main focus on developing economic foundations for a market model of trading private data.

The first part characterizes differential privacy, identifiability and mutual-information privacy by their privacy--distortion functions, which is the optimal achievable privacy level as a function of the maximum allowable distortion. The results show that these notions are fundamentally related and exhibit certain consistency: (1) The gap between the privacy--distortion functions of identifiability and differential privacy is upper bounded by a constant determined by the prior. (2) Identifiability and mutual-information privacy share the same optimal mechanism. (3) The mutual-information optimal mechanism satisfies differential privacy with a level at most a constant away from the optimal level.

The second part studies a market model of trading private data, where a data collector purchases private data from strategic data subjects (individuals) through an incentive mechanism. The value of epsilon units of privacy is measured by the minimum payment such that an individual's equilibrium strategy is to report data in an epsilon-differentially private manner. For the setting with binary private data that represents individuals' knowledge about a common underlying state, asymptotically tight lower and upper bounds on the value of privacy are established as the number of individuals becomes large, and the payment--accuracy tradeoff for learning the state is obtained. The lower bound assures the impossibility of using lower payment to buy epsilon units of privacy, and the upper bound is given by a designed reward mechanism. When the individuals' valuations of privacy are unknown to the data collector, mechanisms with possible negative payments (aiming to penalize individuals with "unacceptably" high privacy valuations) are designed to fulfill the accuracy goal and drive the total payment to zero. For the setting with binary private data following a general joint probability distribution with some symmetry, asymptotically optimal mechanisms are designed in the high data quality regime.

Two thirds of the U.S. power systems are operated under market structures. A good market design should maximize social welfare and give market participants proper incentives to follow market solutions. Pricing schemes play very important roles in market design.

Locational marginal pricing scheme is the core pricing scheme in energy markets. Locational marginal prices are good pricing signals for dispatch marginal costs. However, the locational marginal prices alone are not incentive compatible since energy markets are non-convex markets. Locational marginal prices capture dispatch costs but fail to capture commitment costs such as startup cost, no-load cost, and shutdown cost. As a result, uplift payments are paid to generators in markets in order to provide incentives for generators to follow market solutions. The uplift payments distort pricing signals.

In this thesis, pricing schemes in electric energy markets are studied. In the first part, convex hull pricing scheme is studied and the pricing model is extended with network constraints. The subgradient algorithm is applied to solve the pricing model. In the second part, a stochastic dispatchable pricing model is proposed to better address the non-convexity and uncertainty issues in day-ahead energy markets. In the third part, an energy storage arbitrage model with the current locational marginal price scheme is studied. Numerical test cases are studied to show the arguments in this thesis.

The overall market and pricing scheme design is a very complex problem. This thesis gives a thorough overview of pricing schemes in day-ahead energy markets and addressed several key issues in the markets. New pricing schemes are proposed to improve market efficiency.

Understanding the graphical structure of the electric power system is important

in assessing reliability, robustness, and the risk of failure of operations of this criti-

cal infrastructure network. Statistical graph models of complex networks yield much

insight into the underlying processes that are supported by the network. Such gen-

erative graph models are also capable of generating synthetic graphs representative

of the real network. This is particularly important since the smaller number of tradi-

tionally available test systems, such as the IEEE systems, have been largely deemed

to be insucient for supporting large-scale simulation studies and commercial-grade

algorithm development. Thus, there is a need for statistical generative models of

electric power network that capture both topological and electrical properties of the

network and are scalable.

Generating synthetic network graphs that capture key topological and electrical

characteristics of real-world electric power systems is important in aiding widespread

and accurate analysis of these systems. Classical statistical models of graphs, such as

small-world networks or Erd}os-Renyi graphs, are unable to generate synthetic graphs

that accurately represent the topology of real electric power networks { networks

characterized by highly dense local connectivity and clustering and sparse long-haul

links.

This thesis presents a parametrized model that captures the above-mentioned

unique topological properties of electric power networks. Specically, a new Cluster-

and-Connect model is introduced to generate synthetic graphs using these parameters.

Using a uniform set of metrics proposed in the literature, the accuracy of the proposed

model is evaluated by comparing the synthetic models generated for specic real

electric network graphs. In addition to topological properties, the electrical properties

are captured via line impedances that have been shown to be modeled reliably by well-studied heavy tailed distributions. The details of the research, results obtained and

conclusions drawn are presented in this document.