[Paper Review] Improved Sufficient Conditions for Exact Convex Relaxation of Storage-Concerned ED

Chao Duan, Lin Jiang+3 more|arXiv (Cornell University)|Mar 25, 2016

Electric Power System Optimization4 references3 citations

TL;DR

This paper proposes two improved sufficient conditions for exact convex relaxation in storage-concerned economic dispatch (ED) with complementarity constraints. The first condition, based on Locational Marginal Prices (LMPs), is weaker than prior work, while the second condition ensures exactness when storage energy capacity is sufficiently large at buses where the LMP condition fails. These conditions significantly reduce conservatism in relaxation validity, verified via IEEE 30-bus system tests with up to 10 MWh storage capacity restoring exactness when LMPs drop below the threshold.

ABSTRACT

To avoid simultaneous charging and discharging of storages, complementarity constraints are introduced to storage-concerned economic dispatch (ED), which makes the problem non-convex. This letter concerns the conditions under which the convex relaxation of storage-concerned ED with complementarity constraints is exact. Two new sufficient conditions are proposed, proved and verified to significantly reduce the conservatism of recent results [3], [4].

Motivation & Objective

• To address the challenge of ensuring exact convex relaxation in storage-concerned economic dispatch (ED) with complementarity constraints that prevent simultaneous charging and discharging.
• To reduce the conservatism of existing sufficient conditions for exact relaxation, which often require overly restrictive assumptions on LMPs or storage parameters.
• To propose two new, less conservative sufficient conditions—one based on LMPs and another on storage energy capacity—that guarantee the convex relaxation achieves the same optimal solution as the original non-convex problem.
• To validate the proposed conditions through numerical testing on a real-world IEEE 30-bus system with wind and storage integration.

Proposed method

• Introduces a new sufficient condition (Cond. 1) based on Locational Marginal Prices (LMPs), requiring LMP at each bus and time to exceed a threshold derived from charging cost gradient, discharging cost gradient, and efficiency parameters.
• Proposes a second condition (Cond. 2) that ensures exact relaxation when Cond. 1 fails, provided the storage energy capacity at the violating bus is large enough to prevent overcharging or over-discharging.
• Uses Karush-Kuhn-Tucker (KKT) optimality conditions for the relaxed problem to derive necessary conditions under which simultaneous charging and discharging cannot occur in the optimal solution.
• Applies a duality-based proof strategy: assuming simultaneous charging and discharging in the relaxed solution leads to contradiction if either Cond. 1 or Cond. 2 holds, thus proving the solution lies within the original feasible set.
• Employs a two-step validation procedure: first check Cond. 1 using forecasted LMPs; if violated, estimate required storage capacity and verify Cond. 2.
• Validates the conditions numerically using the SDPT3 solver with YALMIP on a 24-hour, 0.5-hour time-step IEEE 30-bus system with 3 wind farms and 5 energy storage units.

Experimental results

Research questions

• RQ1 Can the sufficient conditions for exact convex relaxation in storage-concerned ED be made less conservative than existing results in [3] and [4]?
• RQ2 Under what conditions on LMPs can the convex relaxation of the non-convex ED problem with complementarity constraints still yield the exact global solution?
• RQ3 How does storage energy capacity influence the validity of convex relaxation when LMP-based conditions are violated?
• RQ4 Can the required storage capacity to restore exactness be estimated based on forecasted LMPs and system parameters?
• RQ5 What is the quantitative improvement in relaxation conservatism when using the proposed conditions compared to prior work?

Key findings

• The proposed LMP-based condition (Cond. 1) is strictly weaker than the condition in [3], reducing the required LMP threshold from 1.5 to -2.76 in the numerical test case.
• When LMPs fall below the threshold in Cond. 1, exact relaxation can still be guaranteed if storage energy capacity is sufficiently large, as formalized in Cond. 2.
• In the numerical test, exactness was lost (simultaneous charging and discharging occurred) when LMPs dropped to -3 and storage capacity was only 2 MWh, with maximum product of charging and discharging power reaching 0.0012.
• After increasing storage capacity to 10 MWh, exactness was restored under the same LMP conditions, with the product of charging and discharging powers reduced to 1.4×10⁻¹², effectively zero.
• The proposed conditions significantly reduce conservatism: Cond. 1 allows valid relaxation at much lower LMPs than [3], and Cond. 2 enables recovery of exactness through larger storage capacity when LMPs are unfavorable.
• The two conditions are complementary: Cond. 1 provides a direct check using forecasted LMPs, while Cond. 2 offers a capacity-based fallback when LMPs are too low.