Regularized MIP Model for Integrating Energy Storage Systems and its Application for Solving a Trilevel Interdiction Problem

TL;DR

This paper introduces a regularized MIP framework for integrating energy storage systems into DCOPF, adding an $\ell_1$-style penalty on charging/discharging to achieve a zero integrality gap between the MIP and its LP relaxation under mild conditions. It proves that, with $E_c^{\min}=E_d^{\min}=0$ and a suitable $\bm\lambda$, the regularized problem matches the LP relaxation closely and yields near-optimal solutions to the original ESS-OPF, while also providing a tight upper bound. The authors further apply this regularized formulation to a three-level $N{-}k$ contingency problem with ESS siting, converting the inner level to an LP via dualization and enabling a tractable bilevel solution with bounded duals and McCormick linearization. Extensive numerical experiments on networks up to 2000 nodes demonstrate small average gaps (often <0.5%) and practical solve times, highlighting the method’s potential for large-scale planning under contingency. The work thus offers a scalable, provably near-optimal approach to ESS-enabled reliability planning in power systems.

Abstract

Incorporating energy storage systems (ESS) into power systems has been studied in many recent works, where binary variables are often introduced to model the complementary nature of battery charging and discharging. A conventional approach for these ESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation models can yield unrealistic fractional solutions, such as simultaneous charging and discharging. In this paper, we develop a regularized Mixed-Integer Programming (MIP) model for the ESS optimal power flow (OPF) problem. We prove that under mild conditions, the proposed regularized model admits a zero integrality gap with its linear programming relaxation; hence, it can be solved efficiently. By studying the properties of the regularized MIP model, we show that its optimal solution is also near-optimal to the original ESS OPF problem, thereby providing a valid and tight upper bound for the ESS OPF problem. The use of the regularized MIP model allows us to solve a trilevel min-max-min network contingency problem which is otherwise intractable to solve.

Figures (4)

• Figure 1: Model comparisons with different choices of $\lambda_c$ and $\lambda_d$.
• Figure 2: Change in the objective value with respect to $\lambda$, where $\lambda_c=\lambda_d=\lambda$ in Proposition \ref{['proposition:best_worst_bound']}. The point denotes the value with $\lambda=(1-\eta^2)/(1+\eta^2)$ for Figures (a), (b), and (c). For Figure (d), the orange dashed line represents the theoretical worst-case bound where the blue solid line shows the empirical difference.
• Figure 3: Long-term planning under $N-k$ contingency problem is a trilevel $\min$-$\max$-$\min$ problem with binary decision variables in each level.
• Figure 4: Benchmark Load Demand $\bm D^0$.

Theorems & Definitions (37)

• Theorem 1
• proof
• Corollary 1
• Example 1
• Example 2
• Remark 1
• Proposition 1
• proof
• Example 3
• Theorem 2