Baosen Zhang

An Optimal and Distributed Method for Voltage Regulation in Power Distribution Systems

This paper addresses the problem of voltage regulation in power distribution networks with deep-penetration of distributed energy resources, e.g., renewable-based generation, and storage-capable loads such as plug-in hybrid electric vehicles. We cast the problem as an optimization program, where the objective is to minimize the losses in the network subject to constraints on bus voltage magnitudes, limits on active and reactive power injections, transmission line thermal limits and losses. We provide sufficient conditions under which the optimization problem can be solved via its convex relaxation. Using data from existing networks, we show that these sufficient conditions are expected to be satisfied by most networks. We also provide an efficient distributed algorithm to solve the problem. The algorithm adheres to a communication topology described by a graph that is the same as the graph that describes the electrical network topology. We illustrate the operation of the algorithm, including its robustness against communication link failures, through several case studies involving 5-, 34-, and 123-bus power distribution systems.

Distributed algorithms for optimal power flow problem

Optimal power flow (OPF) is an important problem for power generation and it is in general non-convex. With the employment of renewable energy, it will be desirable if OPF can be solved very efficiently so that its solution can be used in real time. With some special network structure, e.g. trees, the problem has been shown to have a zero duality gap and the convex dual problem yields the optimal solution. In this paper, we propose a primal and a dual algorithm to coordinate the smaller subproblems decomposed from the convexified OPF. We can arrange the subproblems to be solved sequentially and cumulatively in a central node or solved in parallel in distributed nodes. We test the algorithms on IEEE radial distribution test feeders, some random tree-structured networks, and the IEEE transmission system benchmarks. Simulation results show that the computation time can be improved dramatically with our algorithms over the centralized approach of solving the problem without decomposition, especially in tree-structured problems. The computation time grows linearly with the problem size with the cumulative approach while the distributed one can have size-independent computation time.

Geometry of Power Flows and Optimization in Distribution Networks

We investigate the geometry of injection regions and its relationship to optimization of power flows in tree networks. The injection region is the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region. If the voltage magnitudes are fixed, the injection region of a tree network can be written as a linear transformation of the product of two-bus injection regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Moreover, the resulting convexified optimal power flow problem can be efficiently solved via semi-definite programming or second-order cone relaxations. These results improve upon earlier works by removing the assumptions on active power lower bounds. It is also shown that our practical angle assumption guarantees two other properties: 1) the uniqueness of the solution of the power flow problem and 2) the non-negativity of the locational marginal prices. Partial results are presented for the case when the voltage magnitudes are not fixed but can lie within certain bounds.

Geometry of feasible injection region of power networks

We investigate the problem of power flow and its implications to the optimization in power networks. To understand how to solve these optimization problems, we look at the injection region of power networks. The injection region of a network is the set of all vectors of power injections, one at each bus, that can be achieved while satisfying the network and operation constraints. If there are no operation constraints, we show the injection region of a network is the set of all injections satisfying the conservation of energy. If the network has a tree topology, we show that the injection region with voltage magnitude, line loss constraints, line flow constraints and certain bus power constraints has the same set of Pareto optimal points as its convex hull. The set of Pareto-optimal points are of interest since these are the the optimal solutions to the minimization of a increasing convex function over the injection region. For non-tree networks, we obtain a weaker result by characterize the convex hull of the voltage constraint injection region for lossless cycles, a lossless cycle with a chord and certain combinations of these networks. The convex hull is of interest since they correspond to optimizing linear functions.

Geometry of injection regions of power networks

We investigate the constraints on power flow in networks and its implications to the optimal power flow problem. The constraints are described by the injection region of a network; this is the set of all vectors of power injections, one at each bus, that can be achieved while satisfying the network and operation constraints. If there are no operation constraints, we show the injection region of a network is the set of all injections satisfying the conservation of energy. If the network has a tree topology, e.g., a distribution network, we show that under voltage magnitude, line loss constraints, line flow constraints and certain bus real and reactive power constraints, the injection region and its convex hull have the same Pareto-front. The Pareto-front is of interest since these are the optimal solutions to the minimization of increasing functions over the injection region. For non-tree networks, we obtain a weaker result by characterizing the convex hull of the voltage constraint injection region for lossless cycles and certain combinations of cycles and trees.

Geometry of power flows in tree networks

We investigate the problem of power flow and its relationship to optimization in tree networks. We show that due to the tree topology of the network, the general optimal power flow problem simplifies greatly. Our approach is to look at the injection region of the power network. The injection region is simply the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region, since they are the solutions to the minimization of increasing functions. We view the injection region as a linear transformation of the higher dimensional power flow region, which is the set of all feasible power flows, one for each direction of each line. We show that if the voltage magnitudes are fixed, then the injection region becomes a product of two-bus power flow regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Therefore, the optimal power flow problem can be convexified and efficiently solved. This result improves upon earlier works since it does not make any assumptions about the active bus power constraints. We also obtain some partial results for the variable voltage magnitude case.

Distributed storage for intermittent energy sources: Control design and performance limits

One of the most important challenges in the integration of renewable energy sources into the power grid lies in their ‘intermittent’ nature. The power output of sources like wind and solar varies with time and location due to factors that cannot be controlled by the provider. Two strategies have been proposed to hedge against this variability: 1) use energy storage systems to effectively average the produced power over time; 2) exploit distributed generation to effectively average production over location. We introduce a network model to study the optimal use of storage and transmission resources in the presence of random energy sources. We propose a Linear-Quadratic based methodology to design control strategies, and show that these strategies are asymptotically optimal for some simple network topologies. For these topologies, the dependence of optimal performance on storage and transmission capacity is explicitly quantified.

A local control approach to voltage regulation in distribution networks

This paper address the problem of voltage regulation in power distribution networks with deep penetration of distributed energy resources (DERs) without any explicit communication between the buses in the network. We cast the problem as an optimization problem with the objective of minimizing the distance between the bus voltage magnitudes and some reference voltage profile. We present an iterative algorithm where each bus updates the reactive power injection provided by their DER. The update at a bus only depends on the voltage magnitude at that bus, and for this reason, we call the algorithm a local control algorithm. We provide sufficient conditions that guarantee the convergence of the algorithm and these conditions can be checked a priori for a set of feasible power injections. We also provide necessary conditions establishing that longer and more heavily loaded networks are inherently more difficult to control. We illustrate the operation of the algorithm through case studies involving 8-,34- and 123-bus test distribution systems.

Competition and Coalition Formation of Renewable Power Producers

We investigate group formations and strategic behaviors of renewable power producers in electricity markets. These producers currently bid into the day-ahead market in a conservative fashion because of the real-time risk associated with not meeting their bid amount. It has been suggested in the literature that producers would bid less conservatively if they can form large groups to take advantages of spatial diversity to reduce the uncertainty in their aggregate output. We show that large groups of renewable producers would act strategically to lower the aggregate output because of market power. To maximize renewable power production, we characterize the trade-off between market power and generation uncertainty as a function of the size of the groups. We show there is a sweet spot in the sense that there exists groups that are large enough to achieve the uncertainty reduction of the grand coalition, but are small enough such that they have no significant market power. We consider both independent and correlated forecast errors under a fixed real-time penalty. We also consider a real-time market where both selling and buying of energy are allowed. We validate our claims using PJM and NREL data.

Network Risk Limiting Dispatch: Optimal Control and Price of Uncertainty

Increased uncertainty due to high penetration of renewables imposes significant costs to the system operators. The added costs depend on several factors including market design, performance of renewable generation forecasting and the specific dispatch procedure. Quantifying these costs has been limited to small sample Monte Carlo approaches applied specific dispatch algorithms. The computational complexity and accuracy of these approaches has limited the understanding of tradeoffs between different factors. In this work we consider a two-stage stochastic economic dispatch problem. Our goal is to provide an analytical quantification and an intuitive understanding of the effects of uncertainties and network congestion on the dispatch procedure and the optimal cost. We first consider an uncongested network and calculate the risk limiting dispatch. In addition, we derive the price of uncertainty, a number that characterizes the intrinsic impact of uncertainty on the integration cost of renewables. Then we extend the results to a network where one link can become congested. Under mild conditions, we calculate price of uncertainty even in this case. We show that risk limiting dispatch is given by a set of deterministic equilibrium equations. The dispatch solution yields an important insight: congested links do not create isolated nodes, even in a two-node network. In fact, the network can support backflows in congested links, that are useful to reduce the uncertainty by averaging supply across the network. We demonstrate the performance of our approach in standard IEEE benchmark networks.