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\title{An Online Distributed Optimization Model Solving the Time-Coupled
Conundrum by the Lyapunov Drift Plus Penalty Method}



\author[1]{Molin An}%
\author[1]{Tianguang Lu}%
\author[1]{Xueshan Han}%
\author[2]{Zhaohao Ding}%
\affil[1]{Shandong University}%
\affil[2]{North China Electric Power University}%


\vspace{-1em}



  
  \date{April 25, 2026}


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\begin{abstract}
As the proportion of distributed energy resources (DER) continues to
grow, it is imperative to enhance the economic and safe operation of the
active distribution network (ADN). This paper presents an online
distributed optimization model, in which loads and the active
distribution network (ADN) operators transfer signals and perform local
calculations. In order to significantly reduce the complexity of the
model, a novel linear power flow (LPF) model that can be updated online
is employed. Furthermore, the high proportion of DER access presents a
time-coupled conundrum, thereby creating difficulty in finding the
optimal solution. In this study, the Lyapunov drift plus penalty method
(LDPPM) is employed to address this challenge by transforming the
long-term optimization problem into a current-time one. Additionally, to
enhance the calculation efficiency, the iterative optimization algorithm
is regarded as a control process and is controlled with the PID
controller to form a PID-Lagrange algorithm. The proposed PID-Lagrange
algorithm has a clearly defined parameter tuning strategy that aligns
with control theory, resulting in a significant reduction in operating
costs by 30.27\% in the case study. The results of the performance
analysis demonstrate the superiority of the proposed method.%
\end{abstract}%



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