Optimal Voltage Support and Stress Minimization in Power Networks Marco Todescato, John W. Simpson-Porco, Florian D¨ orfler, Ruggero Carli and Francesco Bullo Abstract—A standard operational requirement in power systems is that the voltage magnitudes must lie within pre- specified bounds. Conventional wisdom suggests that such a tightly regulated voltage profile should also guarantee a secure system, operating far from static bifurcation instabilities such as voltage collapse. Here we demonstrate that this conclusion is generally false, and that the distance from voltage collapse is a systems-level objective distinct from ensuring voltage limits. We formulate an optimization problem which maximizes the distance to voltage collapse through injections of reactive power, subject to power flow and operational voltage constraints. By exploiting a linear reformulation of the power flow equations we arrive at a convex reformulation which can be efficiently solved for the optimal reactive power injections. We illustrate the performance of our results with the IEEE30 bus network. Index Terms— power networks, voltage support, reactive power compensation, sparsity-promoting optimization, optimal placement. I. I NTRODUCTION The widespread penetration of distributed renewable gen- eration, characterized by high variability and fast dynam- ics, negatively impacts the voltage profile of a power net- work. Voltage controllers are therefore required to guarantee constraint satisfaction and safe operation of the network. Techniques for voltage support include shunt and static VAR compensation [1], [2], series compensation [3], off- nominal transformer tap ratios [4], synchronous condensers [5], and more inverters operating away from unity power factor [6], [7]. For a detailed description of the reactive power compensation technologies we refer the reader to [8]. Traditionally, the main purpose of voltage support is to maintain voltage magnitudes tightly within predetermined bounds (e.g., within 5% of some nominal level). Intuitively, such a tightly regulated voltage profile should also guarantee a large stability margin against static bifurcation instabilities such as voltage collapse. A key direction in power system stability analysis has been the development of indices quan- tifying a power network’s proximity to voltage collapse. A broad overview of this large subfield can be found in [9], [10], [11], [12], [13]. The existing approaches are largely based on numerical methods. They often require either continuation power flow [14] to identify the insolvability This work was supported by the Ing. Aldo Gini Foundation, Padova. M. Todescato and R. Carli are with the Department of In- formation Engineering, University of Padova {todescat|carlirug} @dei.unipd.it. J.W. Simpson-Porco and F. Bullo are with the De- partment of Mechanical Engineering and the Center for Control, Dy- namical System and Computation, University of California at Santa Bar- bara {johnwsimpsonporco|bullo} @engineering.ucsb.edu. F. D¨ orfler is with the Automatic Control Laboratory, Swiss Federal Institute (ETH) Zurich dorfler@ethz.ch. boundary, or repeated computation of loading margins in varying directions of parameter-space [15]. In this paper we combine voltage control with metrics quantifying the distance to voltage collapse in order to maximize the networks stability margin. While the usual objective in voltage support problems is the security task of confining voltage magnitudes within predetermined bounds, here we follow an alternative approach: we attempt to minimize a measure of the stress experienced by the network subject to operational constraints. This approach stems from [16], where a condition was introduced which quantifies the network stress measures and the proximity to voltage collapse. Based on this condition we pursue a novel system- level formulation of optimal voltage support encoded as an optimization problem with stress-minimization, i.e., maxi- mization of the distance to voltage collapse, as objective and subject to voltage security constraints. Our decision variables are reactive power injections at a subset of buses equipped with voltage control equipment. This approach allows us to satisfy a local voltage constraint requirements while optimizing a system-level voltage stability margin. By exploiting linear approximation of the power flow equations, we convexify our problem which can then be efficiently solved for the optimal injections. Compared to other approaches to voltage support prob- lems, our results do not rely on the assumption of a radial (i.e., acyclic) power grid topology [6], [7]. This makes our approach appealing for both power transmission and distribution networks. Different from the reactive power compensation literature [17], [6], [18] and from the voltage support literature [7], we seek to maximize a voltage stability margin as opposed to related objectives such as minimizing power losses. The remainder of this paper is organized as follows: in Section II, we recall the required preliminaries. In Section III we present the main contribution of the paper. Firstly, we review the typical objectives for voltage regulation prob- lems and formulate our optimization problem. Secondly, we present and solve the convex reformulation of the problem. In Section IV we present some test cases. Finally, Section V offers concluding remarks. II. PRELIMINARIES A. Power Network, Generator and Load Models A power network can be modelled as a connected, undi- rected and complex-weighted graph G(V , E ) where V = {1,...,n} (|V| = n) represents the set of nodes (or buses), E (|E| = m) is the set of edges (or branches) connecting the nodes, that is the set of unordered pairs (h, k), h,k ∈V ,