Tengmu Li

The feasible region of Alternating Current Optimal Power Flow (ACOPF) models often comprises multiple disjoint feasible components. Each feasible component may contain one or more local optimal solutions (LOSs). Conventional ACOPF solution methodologies typically compute only a single OPF solution and may thereby overlook potential high-value alternative OPF solutions. This paper presents a deterministic numerical method, PGS-TT, to systematically compute all LOSs within each feasible component. PGS-TT integrates the Projected Gradient System with the Trust-Tech framework. This deterministic four-stage method consists of: (1) converging to an LOS; (2) escaping from the current LOS and moving toward another possible OPF solution; (3) detecting the existence of another LOS; and (4) computing the newly detected OPF solution. The proposed numerical PGS-TT method is enhanced by several novel numerical schemes designed to ensure computational efficiency and scalability for large-scale ACOPF models. The method's effectiveness is demonstrated on several modified IEEE test systems: 118-bus, 500-bus, 1,888-bus, and 30,000-bus. Notably, numerical studies reveal several OPF cases where the higher-cost OPF solution exceeds the lower-cost one by up to 51.45%, highlighting the critical need to compute multiple high-quality LOSs to improve the security and economic efficiency of power system operation and planning.

A theory-based Projected Gradient System (PGS)-based Transformation Under Stability-reTaining Equilibria Characterization (Trust-Tech, TT) method, termed PGS-TT, is proposed to systematically and deterministically compute multiple AC Optimal Power Flow (ACOPF) solutions within a feasible component. The theoretical basis for the PGS-TT method is rigorously established, including: (i) a one-to-one correspondence between stable equilibrium points (SEPs) of the PGS and local optimal solutions of ACOPF models; (ii) each feasible component is an invariant set of the PGS and consists of the closures of the stability regions of the SEPs located in that feasible component; (iii) each stability region contains exactly one OPF solution, which eliminates redundant searches for additional OPF solutions within the same stability region; and (iv) the existence conditions of multiple OPF solutions that are linked to the existence of PGS stability boundaries. Theorems and the proposed PGS-TT method are illustrated on 3-bus and 9-bus systems exhibiting multiple local OPF solutions. Additionally, the scalability of the implemented PGS-TT method is discussed based on large-scale systems, with network sizes ranging from 1,888 to 30,000 buses, underscoring its potential for complex power-grid optimization.

Alternating Current Optimal Power Flow (ACOPF) model is nonconvex and generally has multiple local optimal solutions (LOSs) distributed across multiple feasible components of its feasible region. In this paper, a novel, theory-based dynamic method, named QGS-TT, is proposed to compute multiple feasible solutions located in different disconnected feasible components. The proposed method incorporates a dynamic transformation that converts the feasible region of ACOPF problem into the union of regular stable equilibrium manifolds (RSEMs) through the construction of a quotient gradient system (QGS). Furthermore, the Transformation Under Stability Retaining Equilibria Characterization (Trust-Tech, TT) methodology is extended to explore the solution space in a tier-by-tier manner systematically. Starting from an arbitrary initial point, assisted by a QGS-based numerical method to detect the existence of, and then computation of other feasible components, supported by the stability boundary theory (properties). Five numerical studies are conducted on the IEEE 2-, 9-, 39-, 118-, and 500-bus systems, each of which exhibits multiple OPF solutions with significantly different objective function values. Regarding computational performance, it will be shown that the method is significantly faster than an existing one.

This paper reports the second part of a two-part investigation into a novel Augmented Quotient Gradient System (AQGS) method for solving the general AC Optimal Power Flow (OPF) problem. The first part established the theoretical groundwork for the AQGS algorithm, providing formal justification for its advantages. In this second part, we detail the implementation of the robust numerical AQGS algorithm and introduce an enhanced version, referred to as the TJU-enhanced AQGS method, which was designed to improve computational efficiency. Both approaches demonstrate strong convergence, rapid computation, and exceptional robustness, particularly when applied to large-scale OPF problems involving 1888-, 2736-, and 13,695-bus systems. Notably, these methods consistently converge to the optimal solution within a single iteration across multiple test cases.

Due to the non-convex characteristics of the alternating current optimal power flow (ACOPF) problem, most existing methods are restricted to obtaining Karush-Kuhn-Tucker (KKT) solutions. There are significant challenges associated with these solutions, including non-convergence and computational inefficiency. This first part of a two-part paper introduces a robust Augmented Quotient Gradient System (AQGS) method designed to resolve convergence issues in solving the ACOPF problem. The AQGS method demonstrates superior convergence performance, computational efficiency, and a zero-probability occurrence of saddle solutions. A novel dynamical AQGS system is established, then the local optimal solutions of the ACOPF and the regular stable equilibrium points of the AQGS system are shown to correspond one-to-one; the AQGS is demonstrated to be completely theoretically stable. Based on this theoretical foundation, an ideal numerical property is developed. The relevant theorems are visually demonstrated through ACOPF examples, which support the high-performance algorithms introduced in the second part of this two-part paper.

In response to the uncertainties and power quality degradation caused by the integration of numerous renewable energy resources (RESs) and the increasing adoption of plug-in electric vehicles (PEVs) in active distribution networks (ADNs), this paper proposes a three-timescale hierarchical volt/var control (VVC) model operating at 1 hour, 15 minutes, and 1 minute. The model aims to achieve effective coordination between centralized optimization and local control, considering the unique characteristics of different reactive power control devices. A three-stage optimization method is developed for 3-time scale models: for an hourly time scale, a 3-step method is proposed to schedule the tap position of the on-load tap changer (OLTC) and the number of switching groups for the shunt capacitors (SCs). For a 15-minute time scale, the voltage setpoint of the inverter is optimized to minimize the voltage deviations of the pilot buses in typical scenarios, taking into account the uncertainties associated with RESs. For a 1-minute time scale, local measurement information is utilized, and a double-loop droop control method is adopted to track the voltage setting value provided by the 15-minute time scale. This ensures the system’s voltage level and reduces voltage fluctuations. The IEEE 33-bus and IEEE 123-bus distribution networks are employed to validate the proposed method and its ability to deal with uncertainties.