On-Load Tap-Changing Transformers (OLTCs)

• OLTCs are power transformers that adjust discrete tap positions while energized to maintain voltage regulation through deadband-controlled operations.
• They are integral to both transmission and distribution networks, with advanced optimization methods like SOCP and MILP enhancing performance and scalability.
• Coordinated control with inverter-based devices and real-time algorithms has improved renewable hosting capacity and system voltage stability.

An on-load tap-changing transformer (OLTC) is a power transformer equipped with a mechanism that enables discrete adjustment of its turns ratio while energized, thereby providing continuous voltage regulation under load. OLTCs are universally adopted in transmission and distribution networks as the primary voltage-regulating device and their control is essential both for operational security and for accommodating emerging renewable generation and dynamic loads. This article synthesizes the rigorous mathematical formulation, optimization frameworks, dynamic behavior, and the practical system implications of OLTCs as developed in representative research spanning both foundational and state-of-the-art approaches.

1. Functional Principles and Control Logic

The OLTC enables real-time voltage adjustment by selecting among multiple discrete tap positions, each corresponding to a specific winding turns ratio $r = N_p/N_s$. The tap positions are indexed as $k \in \{k_{\min}, ..., k_{\max}\}$ with each step representing a voltage increment or decrement, typically $\Delta V \sim 1$–$2$% per tap. The tap-changing mechanism employs a diverter switch to migrate between taps without interrupting load current, actuated by a motor drive governed by an Automatic Voltage Regulator (AVR). The AVR compares the measured voltage $V_{\mathrm{meas}}$ to a setpoint $V_{\mathrm{ref}}$, only issuing tap commands if $$|V_{\mathrm{meas}} - V_{\mathrm{ref}}| > \Delta V_{\mathrm{db}}$$ (deadband). A mandatory time delay $T_{\mathrm{delay}}$ (e.g., $30$ s) is enforced between operations to prevent excessive mechanical cycling (Zanelli et al., 2 Sep 2025).

Mathematically, tap action is described by first-order lag with deadband: $$\frac{dN}{dt} = \frac{1}{T_{\mathrm{oltc}}}\,\operatorname{sign}(V_{\mathrm{ref}} - V_{\mathrm{meas}})\,u(|V_{\mathrm{ref}} - V_{\mathrm{meas}}| - \Delta V_{\mathrm{db}})$$ where $u[\cdot]$ is the unit step function. Discrete-time update: $$N(k+1) = N(k) + \Delta N \cdot \mathrm{sgn}(V_{\mathrm{ref}}-V_{\mathrm{meas}})$$ applies after each time interval if outside the deadband (Zanelli et al., 2 Sep 2025).

2. OLTC Representation in Network and Optimization Models

The OLTC's discrete tap variable directly modifies the transformer's off-nominal ratio in network admittance matrices, affecting both voltage magnitudes and branch power flows:

• In transmission-level AC-OPF, tap ratios $\tau$ and phase shifts $\theta$ are promoted to optimization variables:
  • Nodal injection: $$S_{\mathrm{bus}} = [V]Y_{\mathrm{bus}}^{*}(\tau,\theta)V^{*}$$
  • Branch admittance: $Y_{ff}=(y+jb + j\,b_c/2)/\tau^2$, $Y_{ft}=-(y+jb)e^{-j\theta}/\tau$ (Bliek, 2013).
• In radial distribution SOCP‐OPF, voltage relations are of the form $U_j = t_{ij}^2 U_i$, where $t_{ij}$ is the discrete tap ratio; bilinearities are exactly linearized via binary expansion and Big-M method, converting the problem into a MISOCP with preserved convexity (Wu et al., 2016).

For multi-period voltage regulation, an affine mapping links discrete tap position $\tau_{p,t}$ to the tap ratio: $$a_{p,t} = 1 + \frac{\tau_{p,t}}{\tau_{p,\mathrm{max}}}(a_{p,\mathrm{max}} - 1)$$ with per-step transition constraints $$|\tau_{p,t} - \tau_{p,t-1}| \leq \Delta TO_{p,\max}$$, ensuring realistic switching speeds (Li et al., 2018, Li et al., 2020, Li et al., 2019).

3. Convexification and Linearization Approaches

Direct modelling of the nonlinear transformer voltage equations and discrete tap changes renders the optimal power flow (OPF) problem nonconvex and computationally prohibitive:

• In SOCP‐OPF, exact linearization is achieved by binary expansion of integer tap variables and Big-M logic for bilinear products (tap binaries $\times$ voltage), converting $U_j = t_{ij}^2 U_i$ into a set of constraints and auxiliary variables; convexity and global optimality are preserved (Wu et al., 2016).
• In AC network models, nonconvex $a^2$ and $\sqrt{\cdot}$ dependencies in voltage computations are handled by first-order Taylor expansion about the current tap setting, rendering all transformer-induced constraints affine in decision variables (Li et al., 2018, Li et al., 2020, Li et al., 2019).
• In large-scale Volt/VAR optimization (VVO), transformer tap ratios $\tau_{ij}$ are discretized (e.g., 33 steps over $[0.9, 1.1]$), incorporated as integer variables in a MINLP; solution proceeds via a relax–round–resolve pipeline: first, relax all integer constraints, solve continuously; second, round to feasible integer settings; third, resolve the fixed-discrete problem for operational feasibility (Tong et al., 29 Jan 2026).

These techniques render coordinated OLTC scheduling computationally tractable on distribution and transmission networks with thousands of buses.

4. Dynamic Behavior, Stability, and System Security

Load tap-changer dynamics interact fundamentally with long-term voltage stability:

• The continuous-time LTC dynamics are

$\dot r_i = \frac{1}{T_i} (V_{s,i}(r) - V_{0,i})$

where $V_{s,i}(r)$ depends on the solved quasi-steady power flow. A unique stable equilibrium exists generically, characterized by rigorous region-of-attraction (ROA) results via QCLP (Cui et al., 2020).

• With predominance of nonlinear loads (e.g., air conditioning units whose $P$ and $Q$ are nonlinear in $V$), conventional OLTC voltage-boosting operations may reduce system power transfer capability or reverse classical stability assignments. Empirical modal analysis confirms nonclassical distribution of stable and unstable equilibria on the $P$–$V$ curve, and tap changes can reduce voltage stability margins, especially under high nonlinear loading (Zanelli et al., 2 Sep 2025).
• New scalable algorithms (SOCP and distributed ADMM) enable online ROA computation and voltage stability assessment under OLTC dynamics, supporting system operator situational awareness and control (Cui et al., 2020).

A robust system model must represent OLTC deadbands, timing delays, discrete steps, and—when present—accurate polynomial $V$–$P$–$Q$ load characteristics to correctly predict voltage stability limits.

5. OLTC Coordination with Emerging Grid Devices

High penetration of distributed generation, especially PV, demands coordination between OLTCs and inverter-based controls:

• Optimization-based coordination frameworks (multi-period convex/MILP) schedule OLTC taps and inverter $Q$ injections to minimize voltage deviations and tap operations, subject to device constraints. These yield reduced voltage unbalance, improved voltage compliance, and strictly controlled tap cycling with computational times compatible with operational practice (e.g., $<1$ s per 30 s time step for 1623-bus, 4-OLTC systems (Li et al., 2018, Li et al., 2020, Li et al., 2019)).
• Proxy-based "soft coordination" (Editor’s term): Instead of centralized control, an actor-critic RL agent dispatches only inverter $Q_{\mathrm{inv}}$, indirectly steering OLTC tap motion by modulating downstream voltage. The inverters never override the OLTC logic; instead, the ensemble minimizes line losses, voltage violations, and tap counts compared to both decentralized Volt–Var droop and batch-scheduled tap trajectories (Wang et al., 2023).
• Empirical studies show that coordinated OLTC–inverter optimization can increase PV hosting capacity by up to $67\%$ over autonomous tap control, reducing voltage excursions and unnecessary tap wear (Li et al., 2018, Li et al., 2020, Li et al., 2019).

6. Practical Impact, Numerical Results, and Guidelines

Quantitative assessments in both transmission and distribution substations confirm the value of optimized OLTC operation:

Study	Network/Device Scale	Solution Time	Voltage Improvement	Tap Operations	Loss/Cost Reduction
(Wu et al., 2016)	IEEE 33 bus, 4 OLTCs	1.6 s	All buses within 0.95–1.05	Few per day	Zero optimality gap; –1.3 kW loss
(Li et al., 2018)	1623 buses, 4 OLTCs	1.1 s per step	Max $\Delta V$ $\downarrow$	$\leq$50/day	Up to $67\%$ more PV hosted
(Li et al., 2020)	2844-node feeder, 4 OLTCs	127 s per step	Max $|V-1|$ $\downarrow$	$\leq$2/day	Line loss and unbalance $\downarrow$
(Tong et al., 29 Jan 2026)	PEGASE 13k buses	$\sim$5 min	MAE$_v$ $\downarrow$, MAE$_q$ $\downarrow$	Discrete (≤33)	Up to $-3.6\%$ gen cost
(Zanelli et al., 2 Sep 2025)	Real voltage collapse event	–	Theory: stability margins	–	Loss of transfer with excessive taps

Implementing OLTC optimization requires accurate device models, including deadband control and tap limits, as well as integration with other voltage-regulating resources. Under high nonlinear loading, conservative tap motion and supervisory controls are advised to prevent counterproductive voltage collapse scenarios (Zanelli et al., 2 Sep 2025).

7. Research Directions and Challenges

State-of-the-art OLTC modeling and optimization frameworks achieve global or near-global optimality, robust voltage regulation, and operational tractability. Yet, practical deployment is contingent on:

• The interplay with heterogeneous and legacy devices, necessitating proxy or indirect coordination methods (soft actor-critic RL, for instance), which avoid hardware upgrades and maintain legacy device autonomy (Wang et al., 2023).
• Robustness under forecast error, grid disturbances, and unmodeled nonlinear loads. Expansion of current convexified formulations to explicitly cover such uncertainties remains an open issue (Li et al., 2018, Li et al., 2020).
• Scalability to full continent-scale systems, with nonconvexities and integer device constraints, solved within operational sub-minute windows (Tong et al., 29 Jan 2026).
• Correctly accounting for the destabilizing potential of OLTC under certain load mixes: OLTC actions can exacerbate rather than mitigate voltage instability if nonlinear loads dominate; real-time monitoring of tap-induced bifurcations and the design of tap supervisory logic are operational imperatives (Zanelli et al., 2 Sep 2025).

OLTCs remain central to voltage regulation in modern power systems. Ongoing research links exact device abstraction, operational optimization, dynamics, and interaction with next-generation grid assets, underlining their enduring technical importance.