IEEE TRANSACTIONS ON 1 Affinely Adjustable Robust Volt/VAr Control without Centralized Computations Firdous U. Nazir, Member, IEEE, Bikash C. Pal, Fellow, IEEE, and Rabih A. Jabr, Fellow, IEEE Abstract—This paper proposes a completely non-centralized Volt/VAr control (VVC) algorithm for active distribution net- works which are faced with voltage magnitude violations due to the high penetration of solar photovoltaics (PVs). The proposed VVC algorithm employs a two-stage architecture where the settings of the classical voltage control devices (VCDs) are decided in the first stage through a distributed optimization engine powered by the alternating direction method of multipliers (ADMM). In contrast, the PV smart inverters are instructed in the second stage through linear Q(P) decision rules - which are computed in a decentralized manner by leveraging robust optimization theory. The key to this non-centralized VVC routine is a proposed network partition methodology (NPM) which uses an electrical distance metric based on node Q -|V | 2 sensitivities for computing an intermediate reduced graph of the network, which is subsequently divided into the final partitions using the spectral clustering technique. As a result, the final network partitions are connected, stable, close in cardinality, contain at least one PV inverter for zonal reactive power support, and are sufficiently decoupled from each other. Numerical results on the UKGDS-95 bus system show that the non-centralized solutions match closely with the centralized robust VVC schemes, thereby significantly reducing the voltage violations compared to the traditional deterministic VVC routines. Index Terms—Network partitioning, Reactive power decision rules, Robust Optimization, Second order conic relaxation, Volt/VAr control (VVC). I. I NTRODUCTION The Volt/VAr control, as part of the distribution management systems, primarily aims at maintaining an acceptable voltage profile throughout the electricity network. In most cases, the node voltage magnitudes should stay between the upper and lower thresholds of 0.95 pu and 1.05 pu. The VVC decides settings for voltage control devices (VCDs) installed on the network through an optimization framework, which requires information about load consumption, PV power generation, network topology, and electrical parameters. The classical VCDs include transformers, on-load tap changers (OLTCs), voltage regulators (VRs), and capacitor banks (CBs), whereas smart PV inverters are also recently being employed in some advanced voltage control routines. Nowadays, the main challenge the traditional deterministic VVC algorithms face is the uncertainty of the PV real power generation. This is because the VVC solutions are dispatched with a typical periodicity of 15-30 minutes, during which the PV real power generation may exhibit enough variations F. U. Nazir and B. C. Pal are with the Electrical and Electronic Engi- neering Department, Imperial College, London SW7 2AZ, U.K. (e-mail: f.ul- nazir16@imperial.ac.uk; b.pal@imperial.ac.uk). R. A. Jabr is with the Department of Electrical and Computer Engineer- ing, American University of Beirut, Beirut 1107 2020, Lebanaon (e-mail: rabih.jabr@aub.edu.lb). to give rise to significant voltage violations [1]. However, simply dispatching the VVC solutions more often is not practically possible owing to the fact that the classical VCDs are switched-type devices which means these are inherently slow in operation, and frequent switching will reduce their life expectancy. In such a situation, reactive power support from PV inverters has been identified as the potential solution by the researchers [2]–[4], given the power electronic nature of the inverters, which enables them to respond almost instantly with negligible wear and tear issues. This fact is also acknowledged by the IEEE 1547-2018 standard, which serves as a specific modification to the original IEEE 1547-2003 standard, to allow dynamic reactive power-based voltage support from the smart inverters [5]. This leads to the problem of the coordinated voltage control between the legacy VCDs and the PV inverters, which has been a subject of recent research. Many research works adopt a control law for adjusting the smart inverter reactive power in response to fluctuations in either the voltage magnitude [6]–[8] or active power [1], [9]– [12] at their terminals, thereby effectively allowing the legacy VCDs to run at a slower time scale. In practice, both the parameters of the control law and the settings of the legacy VCDs are computed simultaneously at the beginning of the optimization horizon. The Q(V ) control law, adopted in [6] - [8], suffers from the main problem of restrictive convergence conditions and can lead to undesirable oscillatory behaviour, whereas the Q(P ) control law has shown stable performance to maintain the voltages in an acceptable region [1], [9]–[12]. Furthermore, the German Grid Code also proposes such Q(P ) characteristics to support the network voltage profile [13]. These Q(P ) characteristics are mostly decided based on affine policies [1], [11], [12] whereas quadratic decision rules have been used in [9], [10]. Although the quadratic decision rules allow finer control, their use is lim- ited by the tractability of the underlying optimization frame- work. As an example, the robust optimization framework of [1], [12] only admits a tractable deterministic counterpart when the control law is determined by affine policies [14]. Such affinely adjustable robust optimization framework has superior performance over other techniques like the chance-constrained optimization, used by [10], [11], in that it guarantees that the voltage violations are completely removed provided the reactive power demanded by the optimized control law does not violate the inverter apparent power limits and the network model is linear. However, one of the main limitations for practical implementation of these robust optimization-based frameworks is that their optimization engine is powered by centralized solvers and thus relies heavily on network ob- servability, remote monitoring, and strong communication in-