Optimal Tap Setting of Voltage Regulation
Transformers in Unbalanced Distribution Systems
Abstract—In this paper, we propose a method to optimally set
the tap position of voltage regulation transformers in distribution
systems. We cast the problem as a rank-constrained semidefinite
program (SDP), in which the transformer tap ratios are captured
by (i) introducing a secondary-side ‘virtual’ bus per transformer,
and (ii) constraining the values that these virtual bus voltages can
take according to the limits on the tap positions. Then, by relaxing
the non-convex rank-1 constraint in the rank-constrained SDP
formulation, one obtains a convex SDP problem. The tap positions
are determined as the ratio between the primary-side bus voltage
and the secondary-side virtual bus voltage that result from the
optimal solution of the relaxed SDP, and then rounded to the
nearest discrete tap values. To efficiently solve the relaxed SDP,
we propose a distributed algorithm based on the Alternating
Direction Method of Multipliers (ADMM). We present several
case studies with single- and three-phase distribution systems to
demonstrate the effectiveness of the distributed ADMM-based
algorithm, and compare its results with centralized solution
methods.
Index Terms—Transformers, Relaxation Methods, Distributed
Algorithms, Decentralized Control, Power System Analysis Com-
puting
I. INTRODUCTION
IN power distribution systems, tap-changing under-load(TCUL) transformers are commonly used for regulating
voltage. Traditionally, automatic voltage regulators (AVRs) are
utilized to control the transformer tap position based on local
voltage measurements (see e.g., [1], [2]). While this AVR-
based control is effective in achieving local voltage regulation,
it is likely not optimal in terms of achieving certain overall
system operational objectives, e.g., minimize power losses and
voltage regulation from some reference value. Motivated by
this, we propose a framework to determine the transformer tap
ratios in distribution systems that is optimal in some sense.
To address the problem described above, we formulate an
optimal power flow (OPF), where the transformer tap ratios are
included as decision variables and the objective is to minimize
the total power losses (although, other objectives can be
accomplished as well). In the context of transmission systems,
optimal transformer tap setting under the OPF framework
has been investigated for decades. For example, in [3], the
transformer tap positions are included as discrete variables in
B. A. Robbins, H. Zhu, and A. D. Domínguez-García are with the
Department of Electrical and Computer Engineering, University of Illi-
nois at Urbana-Champaign, Urbana, IL, 61801. E-mail: {robbins3, haozhu,
aledan}@ILLINOIS.EDU.
This research was supported in part by ABB under project “Distributed and
Resilient Voltage Control of Distributed Energy Resources in the Smart Grid”
(University of Illinois contract UIeRA 2013-2955-00-00), and by the National
Science Foundation (NSF) under grant ECCS-CPS- 1135598 and CAREER
Award ECCS-CAR-0954420.
the OPF problem, which results in a mixed-integer program
(MIP) formulation. Unfortunately, the computational complex-
ity of this formulation grows exponentially as the number
of transformers increases, and thus becomes intractable for
large systems. To tackle this complexity issue, several papers
have proposed to relax transformer tap positions to continuous
optimization variables, and then the solutions to the closest
discrete valuables (see e.g., [3]–[5]). This alternative approach
can yield acceptable performance without incurring the added
complexity. However, all of these approaches are restricted
to standard OPF formulations, and are known to potentially
suffer from the same convergence issues present in traditional
iterative solvers.
In this paper, we formulate the OPF problem that arises
in the context of voltage regulation in distribution systems
as a rank-constrained semidefinite program (SDP), and sub-
sequently obtain a convex SDP problem from the original
SDP formulation by dropping the only non-convex rank-1
constraint (see, e.g., [6]–[9]). In general, this rank relaxation is
not guaranteed to attain the global minimum, in particular for
mesh networks. Interestingly, it has been shown that under
some mild conditions, the optimal solution for the relaxed
SDP-based OPF problem turns out to be of rank 1 for tree-
structured networks, which are typical of radial distribution
systems [6]–[8]. In this sense, the rank relaxation scheme is
actually guaranteed to attain the global optimum of the original
OPF problem. In addition to handling the OPF problem, the
SDP-based approach also constitutes a very promising tool
to tackle the non-convexity in other monitoring and control
applications in power distribution systems.
It is possible to extend the SDP-based OPF approach to
include the tap ratios of TCUL transformers by introducing a
virtual secondary-side bus per transformer, which in turn will
result in additional constraints and decision variables [10]–
[12]. However, the TCUL transformer model proposed in [11]
is limited due to two issues: (i) the relaxed SDP problem could
fail to yield a rank-1 solution, and thus its global optimality is
no longer guaranteed; and (ii) it is only applicable to single-
phase systems. The first issue arises since the network is
equivalently broken into two disconnected parts by introducing
virtual buses associated to each transformer, and the network
disconnection would lead to multiple solutions of rank 2 [12].
Although an optimal rank-1 solution could be recovered in
this case, the conditions for recovering rank-1 solutions are
only possible for single-phase systems [11], [12]. As for
the second issue, it is well known that distribution systems
are unbalanced; this motivates the formulation of the three-