# Modeling and Optimization of Electrical Power Networks

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The electrical power network is moving towards sustainability and cost-efficiency. The former is achieved by incorporating distributed renewable generation and the latter requires employing intelligent control and optimization techniques. As the main interface between the electricity grid and its consumers, the distribution network will play a pivotal role in the near future. In practice, distribution networks operate under unbalanced conditions and far from single-phase simplifications. Thus, the majority of this dissertation aims to provide comprehensive computational tools for multi-phase distribution networks. A final part of this work touches upon transmission networks to explore benefits of combining control and optimization algorithms in various time-scales.

In Chapter 2, models of the most practical distribution elements, including wye and delta ZIP loads, transmission lines with missing phases, step-voltage regulators (SVRs), and three-phase transformers are assembled. Specifically for SVRs, novel nodal admittance models are derived from first principles. Concatenation of these models yields the bus admittance matrix (Y-Bus). Using linear algebra, it is then shown that Y-Bus invertibility is compromised only when the network includes ungrounded or delta-connected transformers. For such devices, we mathematically show why a previously proposed modification in their nodal admittance restores Y-Bus invertibility. Mathematical guarantees for Y-Bus invertibility is important since, for instance, it allows one to run the Z-Bus method to compute voltage solutions of power flow equations.

In Chapter 3, theoretical convergence of the Z-Bus method in multi-phase distribution networks with wye and delta ZIP loads is studied. By viewing the Z-Bus method as a fixed-point iteration, sufficient conditions for its contraction are derived. These conditions define a region, expressed in terms of Y-Bus and ZIP loads, in which unique voltage solutions to power flow equations exist.

Chapter 4 considers a planning problem for inverter-based renewable systems in multi-phase distribution networks. The objective is to minimize the installation costs of distributed generators (DG) during the planning stage and the costs of power import plus DG curtailment during operations. Three- and single-phase inverter models that preserve the underlying mapping between renewable uncertainty to power injection are presented. Scenario-based characterization of distributed generation and loads as well as power flow linearizations are leveraged to render a stochastic formulation for optimal DG placement and sizing. The proposed problem is a mixed-integer second-order cone program that is solved efficiently. Simulations on several medium-to-large-sized distribution test feeders promise that optimal stochastic planning of DGs reduces costs during validation, compared to a scheme where uncertainty is only represented by its average value.

Chapter 5 presents an optimal power flow (OPF) problem that allows for tap selection of various types of SVRs. The goal is to minimize power import while satisfying operational constraints. A set of power flow equations are derived that explicitly account for the tap ratios based on the nodal admittance model of SVRs (Chapter 2). Chordal semidefinite relaxations of the power flow equations are pursued for non-SVR edges. For each SVR type, novel relaxations are proposed to handle the non-convex primary-to-secondary voltage relationship. The formulation is a semidefinite program (SDP). Numerical tests on the IEEE 37-bus distribution feeder indicate the success of the proposed SDP in selecting taps of wye, closed-delta, and open-delta SVRs.

Chapter 6 augments the transmission OPF problem with a load-following controller whose costs are expressed through the linear quadratic regulator (LQR). The power network is described by a set of nonlinear differential algebraic equations (DAEs). By linearizing the DAEs around a known equilibrium, a linearized OPF with operational constraints is formulated first. This OPF is then augmented by a set of linear matrix inequalities equivalent to the implementation of an LQR controller. The resulting formulation, termed LQR-OPF, is an SDP which furnishes optimal steady-state setpoints and an optimal feedback law to steer the system to the new steady state with minimum load-following control costs. Experiments on test cases demonstrate that the setpoints computed by LQR-OPF result in lower overall costs and frequency deviations compared to those of a scheme where OPF and load-following control are considered separately.