Voltage Regulation Algorithms for Multiphase Power Distribution Grids Vassilis Kekatos, Member, IEEE,Liangzhang, Student Member, IEEE,Georgios B
Total Page：16
File Type：pdf, Size：1020Kb
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 31, NO. 5, SEPTEMBER 2016 3913 Voltage Regulation Algorithms for Multiphase Power Distribution Grids Vassilis Kekatos, Member, IEEE,LiangZhang, Student Member, IEEE,Georgios B. Giannakis, Fellow, IEEE, and Ross Baldick, Fellow, IEEE Abstract—Time-varying renewable energy generation can result conditions requires advanced reactive power management in serious under-/over-voltage conditions in future distribution solutions. To that end, the power electronics of PV inverters grids. Augmenting conventional utility-owned voltage regulating and storage devices offer a decentralized and fast-responding equipment with the reactive power capabilities of distributed gen- eration units is a viable solution. Local control options attaining alternative [27], [29]. global voltage regulation optimality at fast convergence rates is the A grid operator can engage the reactive power capabili- goal here. In this context, novel reactive power control rules are ties of distributed generation (DG) units to minimize power analyzed under a unifying linearized grid model. For single-phase losses while satisfying voltage regulation constraints. Being an grids, our proximal gradient scheme has computational com- instance of the optimal power flow (OPF) problem, reactive plexity comparable to that of the rule suggested by the IEEE 1547.8 standard, but it enjoys well-characterized convergence power support can be solved using convex relaxation tech- guarantees. Furthermore, adding memory to the scheme results niques [12], [19]. Among other centralized approaches, inverter in accelerated convergence. For three-phase grids, it is shown that VAR control is solved using convex relaxation in [12], while a reactive power injections have a counter-intuitive effect on bus scheme relying on successive convex approximation is devised voltage magnitudes across phases. Nevertheless, when our control in [11]. Distributed algorithms requiring communication across scheme is applied to unbalanced conditions, it is shown to reach an equilibrium point. Numerical tests using the IEEE 13-bus, the nodes have been proposed too. A distributed method based on IEEE 123-bus, and a Southern California Edison 47-bus feeder convex relaxation has been developed in [31]. Upon modeling with increased renewable penetration verify the properties of the power losses as a quadratic function of reactive power injec- schemes and their resiliency to grid topology reconfigurations. tions, [8] pursues a consensus-type algorithm. Control rules Index Terms—Accelerated proximal gradient, linear distribu- based on approximate models are presented in [29], and [2] tion flow model, PV inverters, three-phase distribution grids. developes a multi-agent scheme. Building on the radial struc- ture of distribution grids, algorithms requiring communication only between adjacent nodes have been developed based on the I. INTRODUCTION alternating-direction method of multipliers (ADMM) [6], [25], [28]. OLTAGE regulation, that is the task of maintaining bus To cater the scalability of DG units and the potential lack V voltage magnitudes within desirable levels, is critically of communication infrastructure, local plug-and-play schemes challenged in modern distribution grids. The penetration of are highly desirable. Given that voltage magnitudes depend on renewables, demand-response programs, and electric vehicles grid-wide reactive injections, guaranteeing voltage regulation lead to time-varying active power injections and frequently constraints may be hard to accomplish via purely localized al- reversing power flows. Utility-owned equipment convention- gorithms. In such setups, reactive power management is usu- ally employed for voltage regulation, such as tap-changing ally relaxed to penalizing voltage magnitude deviations from transformers and shunt capacitors, cannot react promptly and ef- a desired value and neglecting power losses. Reactive power ficiently enough [1], [10]. Hence, avoiding under-/over-voltage injections are adjusted proportionally to the local voltage vio- lations in [26]; see also [32] for sufficient conditions guaran- teeing its convergence. A similar local control strategy has been Manuscript received May 06, 2015; revised May 13, 2015 and August 27, showntominimizeamodifiedvoltageregulationcostin[14], 2015; accepted October 16, 2015. Date of publication November 03, 2015; date while [13] proposes a subgradient-based algorithm. A control of current version August 17, 2016. This work was supported by the NSF under grants 1423316, 1442686, 1508993, and 1509040. Paper no. TPWRS-00633- rule adjusting the inverter voltage output according to the re- 2015. active power flow is reported in [17]. The local control rules V. Kekatos is with the Electrical and Computer Engineering Department, Vir- proposed in [22] maintain voltage magnitudes within the de- ginia Tech, Blacksburg, VA 24060 USA (e-mail: [email protected]). L. Zhang and G. B. Giannakis are with the Digital Technology Center and sired range under the presumption of unlimited reactive power the Electrical and Computer Engineering Department, University of Minnesota, support. Minneapolis, MN 55455 USA (e-mail: [email protected]; georgios@umn. Most existing works build on a simplified single-phase grid edu). R. Baldick is with the Electrical and Computer Engineering Department, Uni- model. Due to untransposed distribution lines and unbalanced versity of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]. loads though, the equivalent single-phase distribution network edu). may not exist. Semidefinite programming and ADMM-based Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. schemes have been applied in multiphase radial networks for Digital Object Identifier 10.1109/TPWRS.2015.2493520 power flow optimization [33], [16]. Nonetheless, no work dis- 0885-8950 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 3914 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 31, NO. 5, SEPTEMBER 2016 closes how inter-phase coupling affects bus voltage magnitudes nodes can be numbered such that for all .For across phases, or how local control schemes behave in unbal- every bus ,let be its squared voltage magnitude, and anced grids. its complex power injection. This work considers local reactive power control rules for The distribution line connecting bus with its parent is voltage regulation in single- and multi-phase distribution grids. indexed by . For every line ,let , , Our contribution is on four fronts. First, we provide a unified and be the line impedance, the squared current matrix-vector notation for approximate yet quite accurate magnitude, and the complex power flow sent from the sending multi-phase grid models (Section II). Second, Section III ex- bus , respectively. If is the set of children buses for bus , tends the work of [20]. In [20], we developed a reactive the grid can be modeled by the branch flow model [3], [4] power control rule based on a proximal gradient scheme, and engineered modifications with superior convergence prop- (1a) erties. Here, the options of unlimited reactive support and diagonal scaling are considered too. In particular, numerical (1b) results indicate that the convergence rates attained by using (1c) different step sizes across buses are still significantly lower than those achieved by our accelerated scheme. Third, using for all , and the initial condition . a linear approximation for unbalanced multi-phase grids, we For notational brevity, collect all nodal quantities re- reveal an interesting inter-phase coupling pattern across buses lated to non-feeder buses in vectors , (Section IV). Recall that in single-phase grids, increasing the ,and . Similarly for lines, reactive power injection at any bus raises the nodal voltage introduce vectors , , magnitudes throughout the grid. In multi-phase grids on the ,and . Define further the contrary, injecting more reactive power into a bus of one phase complex vectors , ,and . could result in decreasing voltage magnitudes for the preceding According to the approximate LinDistFlow model, the grid is in the positive-sequence ordering phase. It is finally shown described by the linear equations [3], [4] that in unbalanced scenarios our reactive power control rule converges to an equilibrium point; yet this point does not (2a) necessarily correspond to the minimizer of a voltage regulation (2b) problem. Numerical tests on distribution feeders corroborate the convergence properties of the novel schemes, as well as where is the squared voltage magnitude at the feeder; matrix their resiliency to topology reconfigurations (Section V). is defined as ;and is the reduced branch-bus Regarding notation, lower- (upper-) case boldface letters de- incidence matrix enjoying the following properties. note column vectors (matrices), with the exception of line power Proposition 1 ([20]): The negative of the reduced branch-bus flow vectors . Calligraphic symbols are reserved for sets. incidence matrix and its inverse satisfy: Symbol stands for transposition. Vectors , ,and ,arethe (p1) they are both lower triangular with unit eigenvalues; all-zeros, all-ones, and the -th canonical vectors, respectively. (p2) ;and Symbol denotes the -norm of vector , while where is the full branch-bus incidence matrix. stands for the -th largest eigenvalue of .Operator Equation