Mohammadhafez Bazrafshan

Decentralized stochastic programming for real and reactive power management in distribution systems

The stochastic nature of solar renewable power poses challenges in distribution networks with high-penetration photovoltaic (PV) generation in terms of achieving thermal loss minimization, voltage regulation, and customer satisfaction. This paper introduces a stochastic optimization model for real and reactive power management in such distribution systems with high level of residential-scale PV penetration. Decision variables include demand response schedules of programmable loads, as well as reactive power consumption or generation by the PV inverters in a fashion adaptive to the uncertain real power generated by the PV units. A decentralized algorithm based on the alternating direction method of multipliers (ADMM) is developed. The algorithm features closed-form updates per node, and requires communication only between neighboring nodes. The performance of the algorithm is evaluated on a typical rural distribution circuit using computer simulations. The proposed method is demonstrated to have superior voltage regulation in comparison to local reactive power control alternatives.

Decentralized Stochastic Optimal Power Flow in Radial Networks With Distributed Generation

This paper develops a power management scheme that jointly optimizes the real power consumption of programmable loads and reactive power outputs of photovoltaic (PV) inverters in distribution networks. The premise is to determine the optimal demand response schedule that accounts for the stochastic availability of solar power, as well as to control the reactive power generation or consumption of PV inverters adaptively to the real power injections of all PV units. These uncertain real power injections by PV units are modeled as random variables taking values from a finite number of possible scenarios. Through the use of second order cone relaxation of the power flow equations, a convex stochastic program is formulated. The objectives are to minimize the negative user utility, cost of power provision, and thermal losses, while constraining voltages to remain within specified levels. To find the global optimum point, a decentralized algorithm is developed via the alternating direction method of multipliers that results in closed-form updates per node and per scenario, rendering it suitable to implement in distribution networks with a large number of scenarios. Numerical tests and comparisons with an alternative deterministic approach are provided for typical residential distribution networks that confirm the efficiency of the algorithm.

Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

This paper derives a set of sufficient conditions guaranteeing that the load-flow problem in unbalanced three-phase distribution networks with wye and delta constant-power, constant-current, and constant-impedance loads (ZIP loads) has a unique solution over a region that can be explicitly calculated from the network parameters. It is also proved that the well-known Z-Bus iterative method is a contraction over the defined region, and hence converges to the unique solution.

Placing and sizing distributed photovoltaic generators for optimal reactive power compensation

A two-stage stochastic programming approach is pursued to optimally place and size photovoltaic (PV) inverters in a radial distribution network under solar irradiance and load uncertainties. First-stage variables include binary PV unit placement as well as continuous real and apparent power capacities of the inverters. Second-stage decisions comprise reactive power compensation, power flows, and nodal voltages, which are determined adaptive to the uncertainty. The objective is to minimize installation cost and expected thermal losses on the network. The optimization problem is formulated as a mixed-integer second-order cone program and is tested on a practical distribution network.

Comprehensive Modeling of Three-Phase Distribution Systems via the Bus Admittance Matrix

The theme of this paper is three-phase distribution system modeling suitable for the Z-Bus load-flow. Detailed models of wye and delta constant-power, constant-current, and constant-impedance loads are presented. Models of transmission lines, step-voltage regulators, and transformers that build the bus admittance matrix (Y-Bus) are laid out. The Z-Bus load-flow is then reviewed and the singularity of the Y-Bus in case of certain transformer connections is rigorously discussed. Based on realistic assumptions and conventional modifications, the invertibility of the Y-Bus is proved. Last but not least, MATLAB scripts that model the components of the IEEE 37-bus, the IEEE 123-bus, the 8500-node feeders, and the European 906-bus low-voltage feeder are provided.

Convergence of the Z-Bus method and existence of unique solution in single-phase distribution load-flow

For a single-phase distribution network with constant-power, constant-current, and constant-impedance loads (ZIP loads), sufficient conditions are presented that explicitly define a region where a unique load-flow solution exists. The Z-Bus method is shown to be a contraction mapping iteration, which upon initialization within this region, is guaranteed to converge to the unique load-flow solution. The sufficient conditions for convergence of the Z-Bus method are numerically verified for IEEE distribution test feeders.

Augmenting the optimal power flow for stability

This paper presents an augmented optimal power flow (OPF) formulation that minimizes a power networks transient control costs using a linear quadratic regulator (LQR). The network is described by AC power flows with third-order generator dynamics modeling. Then, linearized dynamics around a known solution of the power flow equations are considered. Leveraging the equivalent linear matrix inequality formulation for the LQR, the augmented OPF (LQR-OPF) amounts to a semidefinite program, yielding optimal network steady state and an explicit feedback gain for minimum transient control cost. Numerical tests on a standard power network demonstrate the advantage of LQR-OPF in comparison to a scheme where OPF and transient control are solved separately.

On the solution of the three-phase load-flow in distribution networks

This paper is concerned with the Z-Bus method to solve the load-flow problem in three-phase distribution networks with wye and delta constant-power, constant-current, and constant-impedance loads (ZIP loads). The Z-Bus method is viewed as a fixed-point iteration. By leveraging the contraction mapping theorem, a set of sufficient conditions is then presented that guarantees a) the existence of a unique solution over a region that can be computed from the network parameters, and b) the convergence of the Z-Bus method to the unique solution. It is numerically illustrated that the new set of sufficient conditions holds for practical distribution networks and improves the previously reported results on the convergence of the Z-Bus method in three-phase networks.

Voltage regulation in electricity distribution networks using the conditional value-at-risk

Voltage regulation in distribution networks featuring high penetration of distributed photovoltaic (PV) generation is particularly challenging due to the stochastic nature of solar energy. To ensure that voltage levels remain within safety margins, this paper introduces a real and reactive power optimization model that penalizes the conditional value-at-risk of the voltage deviation from its nominal value. This risk-averse approach guarantees that node voltages across the network remain close to their nominal value with a specified probability. Decision variables are the real power consumption of controllable loads and reactive power consumption or generation from PV inverters. Adopting a scenario-based model for the uncertain solar power, the overall problem amounts to a convex quadratic program. Numerical tests for typical residential distribution networks exhibit the effectiveness of the model in achieving voltage regulation as compared to alternative risk-neutral approaches.

Coupling Load-Following Control with OPF

In this paper, the optimal power flow (OPF) problem is augmented to account for the costs associated with the load-following control of a power network. Load-following control 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 that accounts for steady-state operational constraints is formulated first. This linearized OPF is then augmented by a set of linear matrix inequalities that are algebraically equivalent to the implementation of an LQR controller. The resulting formulation, termed LQR-OPF, is a semidefinite program 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. Numerical tests demonstrate that the setpoints computed by LQR-OPF result in lower overall costs and frequency deviations compared to the setpoints of a scheme where OPF and load-following control are considered separately.