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When are Lossy Energy Storage Optimization Models Convex?

Feras Al Taha, Eilyan Bitar

Abstract

We examine a class of optimization problems involving the optimal operation of a single lossy energy storage system, where energy losses occur during charging and discharging. These inefficiencies typically lead to a nonconvex set of feasible charging and discharging power profiles. In this paper, we derive an equivalent reformulation of this class of optimization problems by eliminating the charging and discharging power variables and recasting the problem entirely in terms of the storage state-of-charge variables. We show that the feasible set of the proposed reformulation is always convex. We also provide sufficient conditions under which the objective function of the proposed reformulation is guaranteed to be convex. The conditions provided both unify and generalize many existing conditions for convexity in the literature.

When are Lossy Energy Storage Optimization Models Convex?

Abstract

We examine a class of optimization problems involving the optimal operation of a single lossy energy storage system, where energy losses occur during charging and discharging. These inefficiencies typically lead to a nonconvex set of feasible charging and discharging power profiles. In this paper, we derive an equivalent reformulation of this class of optimization problems by eliminating the charging and discharging power variables and recasting the problem entirely in terms of the storage state-of-charge variables. We show that the feasible set of the proposed reformulation is always convex. We also provide sufficient conditions under which the objective function of the proposed reformulation is guaranteed to be convex. The conditions provided both unify and generalize many existing conditions for convexity in the literature.
Paper Structure (6 sections, 3 theorems, 15 equations, 1 figure, 1 table)

This paper contains 6 sections, 3 theorems, 15 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Problem Formulation
  4. Convex Reformulation
  5. Discussion
  6. Conclusion

Key Result

Theorem 1

The function $\phi: \mathbb{R}^T \rightarrow \mathbb{R}^T$, as defined in Eq. eq:phi_map, is concave and invertible.When referring to convexity (resp. concavity) of a vector-valued function, we mean that each of its scalar-valued component functions is a convex (resp. concave) function. Furthermore, for all $x \in \mathbb{R}^T$.

Figures (1)

  • Figure 1: Depiction of (a) the set of feasible power profiles $\mathbb{U}$ and (b) the set of feasible energy profiles $\mathbb{X}=\phi(\mathbb{U})$ for a simple two-period ($T=2$) lossy energy storage system model with parameters: $\eta^{c}=\eta^{ d}=0.5$, $\Delta=1$, $x_0=0.75$, $\lambda=1$, $\underline{u}=(1,1)$, $\overline{u}=(1,1)$, $\underline{x}=(0,0)$, and $\overline{x}=(1,1)$.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Theorem 3
  • proof