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Stochastic Operation of Distribution Networks with Voltage Dependent Load



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Uncertainties in load and renewable generations impose new challenges on the operation of distribution networks due to the complexity of physical models and the randomness at the distribution level. This topic has been increasingly studied due to the growing accessibility of real-time data at the distribution level and the increasing uncertainties from high-penetration of renewable generations and the integration of electric vehicles. This work starts with the formulation of adequate network and load models in a deterministic problem. Then the uncertainties are included in these models to find the probability distribution function (PDF) of the state of the system in a probabilistic load flow (PLF). The uncertainty model of the PLF is later used to formulate an optimal power flow (OPF) to find the optimal input of the controllable sources in a typical distribution network. Finally, battery energy storage systems (BESS) are added to the OPF problem, which is further generalized through a distributionally robust approach by considering more realistic randomness with an uncertain PDF. The oncoming permanently close operation of normal open switches between feeders in distribution systems results in larger and meshed networks, which imposes challenges for the real-time operation now required for the supervisory control and data acquisition. This has accelerated the development of decentralized and fast tools for the analysis of new larger distribution networks. In this work, a three-phase decentralized load flow method is developed for distribution networks based on a novel extended current injection model (CIM) with the voltage dependence of the load expressed by the constant-impedance, constant-current and constant-power (ZIP) load model. A hybrid method that combines the good behavior of the Newton-Raphson method over constant power loads and the fixed-point method over constant current load is presented to solve the proposed extended CIM. To consider unbalanced uncertainties from voltage-sensitive loads and photovoltaic (PV) generation in distribution networks, this work proposes a fully analytic second-order probabilistic load flow (PLF) method to realize an accurate and fast three-phase load flow analysis based on the bus injection model (BIM), which is a derivation of the CIM. The load flow equations are modeled using an accurate quadratic expression. To work at the distribution level, the voltage dependence of the load is considered. The uncertainties are modeled in time series as conditional probabilities, reducing the complexity of their PDFs. The PLF is modeled in a fully analytic second-order second-moment (SOSM) stochastic formulation, which can accurately and easily handle PDFs of voltage and current by computing the first two moments. The computation is accelerated by an analytical calculation of the quadratic coefficients over the ZIP parameters. With the proposed SOSM model, this work develops a stochastic AC optimal power flow (SOPF) method. It considers the load voltage dependence, unbalance, and correlation that exist in distribution systems. The proposed SOPF handles the uncertainties of PV sources and loads as voltage sensitive parameters, caused by communication delays. The SOPF method optimizes the active and reactive power of BESS in a static model, as well as the reactive power of PV generators and static var compensators (SVCs). As ZIP parameters are random variables in the PLF, the accurate SOSM model is applied to map the complex uncertainties over the complex voltage to handle the strongly nonlinear relation between the voltage and random ZIP parameters. A second-moment chance constraint is formulated to tackle the voltage and current magnitude over the nonlinear SOSM, based on a conic generalization of the Chebyshev bound. The resulting constraint handles the joint violation considering the correlation among individual constraints and also relaxes the nonconvex minimum voltage constraint. With the inclusion of BESS in the distribution networks, the SOPF problem has to deal with energy horizon constraints, which turns the problem into a multistage OPF problem under uncertainties. Tackling this problem at the distribution level implies dealing with load voltage dependence, which complicates the models. Furthermore, the load behavior at the distribution level changes over time, which leads to a changing PDF. To obtain a good balance between handling the risk of the highly changing load and optimality, the distributionally robust (DR) approach is applied. A distributionally robust multistage OPF (DR-MOPF) is proposed to deal with data-based formulations which rapidly grow in size with the amount of information available. The proposed new paradigm for DR-OPF leverages the optimality advantages of the DR methods and the speed advantages of a conic ambiguity set made around the first two moments of the load and stochastic generation. The optimization minimizes the multistage loss risk which is conic-representable based on the nested property of the expectation. Constraint risk violations being expressed as a distributionally robust CVaR are also conic representable. It is a data-driven approach that learns from data without growing with them.



Stochastic optimization, Distributionally robust, Conic optimization, Load model.



Doctor of Philosophy (Ph.D.)


Electrical and Computer Engineering


Electrical Engineering


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