Variance-Aware Optimal Power Flow

Abstract

The incorporation of stochastic loads and generation into the operation of power grids gives rise to an exposure to stochastic risk. This risk has been addressed in prior work through a variety of mechanisms, such as scenario generation or chance constraints, that can be incorporated into OPF computations. Nevertheless, numerical experiments reveal that the resulting operational decisions can produce power flows with very high variance. In this paper we introduce a variety of convex variants of OPF that explicitly address the interplay of (power flow) variance with cost minimization, and present numerical experiments that highlight our contributions.

Variance-Aware Optimal Power Flow: Addressing the Tradeoff Between Cost, Security, and Variability

“…This paper continues the work initiated in [18]. We describe modifications to the OPF problem which we term varianceaware OPF.…”

“…• Line safety parameters are set to 3. It can again be easily shown [18] that the solution given by ( 16) remains feasible, and is the sole optimal solution. The variance metric (17) attained by solution ( 16) equals 1/81 + 1/(4k) ≈ .0123 for k large.…”

“…As outlined above in the Gaussian case ( 12) is equivalent to (9) when ν ij = Φ −1 (1− ). However, the Gaussian case is not the only one where such an equivalence holds; see the discussion in [18]. In each case one needs an appropriately constructed parameter ν ij = ν ij ( ).…”

“…is optimal (in fact: uniquely optimal) for the case of problem (15) in Figure 1. For a proof, see [18].…”

“…As the reader can observe, one can scale the ∆ term in (18a) by a positive constant, obtaining a problem with the same functional form that places a different emphasis on the costvariance trade-off. Seen from this perspective, problem (18) bears a resemblance to traditional mean-variance optimization problems arising in financial portfolio analysis; see [23]. In Section VI we will return to this viewpoint.…”