Risk-Averse Optimization and its Applications in Power Grids with Renewable Energy Integration

Abstract

Electric power is one of the most critical parts of everyday life; from lighting, heating, and cooling homes to powering televisions and computers. The modern power grids face several challenges such as efficiency, sustainability, and reliability. Increase in electrical energy demand, distributed generations, integration of uncertain renewable energy resources, and demand side management are among the main underlying reasons of such growing complexity. Additionally, the elements of power systems are often vulnerable to failures because of many reasons, such as system limits, poor maintenance, human errors, terrorist/cyber attacks, and natural phenomena. One common factor complicating the operation of electrical power systems is the underlying uncertainties from the demands, supplies and failures of system components. Stochastic optimization approaches provide mathematical frameworks for decision making under uncertainty. It enables a decision maker to incorporate some knowledge of the uncertainty into the decision making process to find an optimal trade off between cost and risk. In this dissertation, we focus on application of three risk-averse approaches to power systems modeling and optimization. Particularly, we develop models and algorithms addressing the cost-effectiveness and reliability issues in power grids with integrations of renewable energy resources. First, we consider a unit commitment problem for centralized hydrothermal systems where we study improving reliability of such systems under water inflow uncertainty. We present a two-stage robust mixed-integer model to find optimal unit commitment and economic dispatch decisions against extreme weather conditions such as drought years. Further, we employ time series analysis (specifically vector autoregressive models) to construct physical based uncertainty sets for water inflow into the reservoirs. Since extensive formulation is impractical to solve for moderate size networks we develop an efficient Benders' decomposition algorithm to solve this problem. We present the numerical results on real-life case study showing the effectiveness of the model and the proposed solution method. Next, we address the cost effectiveness and reliability issues considering the integration of solar energy in distributed (decentralized) generation (DG) such as microgrids. In particular, we consider optimal placement and sizing of DG units as well as long term generation planning to efficiently balance electric power demand and supply. However, the intermittent nature of renewable energy resources such as solar irradiance imposes several difficulties in decision making process. We propose two-stage stochastic programming model with chance constraints to control the risk of load shedding (i.e., power shortage) in distributed generation. We take advantage of another time series modeling approach known as autoregressive integrated moving average (ARIMA) model to characterize the uncertain solar irradiance more accurately. Additionally, we develop a combined sample average approximation (SAA) and linearization techniques to solve the problem more efficiently. We examine the proposed framework with numerical tests on a radial network in Arizona. Lastly, we address the robustness of strategic networks including power grids and airports in general. One of the key robustness requirements is the connectivity between each pair of nodes through a sufficiently short path, which makes a network cluster more robust with respect to potential disruptions such as man-made or natural disasters. If one can reinforce the network components against future threats, the goal is to determine optimal reinforcements that would yield a cluster with minimum risk of disruptions. We propose a risk-averse model where clusters represents a R-robust 2-club, which by definition is a subgraph with at least R node/edge disjoint paths connecting each pair of nodes, where each path consists of at most 2 edges. And, develop a combinatorial branch-and-bound algorithm to compare with an equivalent mathematical programming approach on random and real-world networks.