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Operational performance in the low-voltage distribution network (LVDN) can be undermined by its inherent unbalances, which may become worse as the penetration of rooftop solar continuously increases. To address this issue, load balancing via phase-reconfiguration devices (PRDs), which can change phase positions of residential customers as required, provides a cost-efficient option. However, most reported approaches to control PRDs require that demands of all residential customers are available, which are not viable for many LVDNs without smart meters or advanced metering infrastructure (AMI) installed. To bridging the gap in this field, this paper proposes a novel method to control PRDs purely based on measurable data from PRDs, and its controller. Based on limited information, sensitivity analysis in the network with PRDs is studied, followed by the optimization model that comprehensively considers operational requirements in the network. Moreover, slack variables are introduced to the model, and penalized in the objective function to assure either a strategy that is secure or with minimized violations can always be provided. The model is a challenging mixed-integer non-convex programming (MINCP) problem, which is reformulated as an efficient solvable mixed-integer second-order cone programming (MISOCP) based on exact reformulations or accurate linear approximations. Simulations based on two modified IEEE systems, and a real system in Australia demonstrate that an efficient strategy can be provided to mitigate unbalances in the network. [ABSTRACT FROM AUTHOR]