A two-stage non-linear program for optimal electrical grid power balance under uncertainty

Abstract

We propose a two-stage non-linear stochastic formulation for the economic dispatch problem under renewable-generation uncertainty. Each stage models dispatching and transmission decisions that are made on subsequent time periods. Certain generation decisions are made only in the first stage and the second stage realizes the actual renewable generation, where the uncertainty in renewable output is captured by a finite number of scenarios. Any resulting supply-demand mis-match must then be alleviated using extra, high marginal-cost power sources that can be tapped in short order. We propose two outer approximation algorithms to solve this nonconvex optimization problem to optimality. We show that under certain conditions the sequence of optimal solutions obtained under both alternatives has a limit point that is a globally-optimal solution to the original two-stage nonconvex program. Numerical experiments for a variety of parameter settings were carried out to indicate the efficiency and usability of this method of large practical instances.

Physics of power networks makes hard optimization problems easy to solve

“…To substantiate that most of the 12 controllable phase shifters are not important, notice that • OPF with a single variable phase shifter on the line (25,27) has the optimal cost 573.92, corresponding to the phase 7.82 • . • OPF with two variable phase shifters on the lines (25,27) and (8,28) gives the optimal cost 573.67, corresponding to the optimal phases 5.70 • and −0.30 • . We have repeated the above experiment for several random cost functions and observed that OPF with 1-2 phase shifters is a good approximation of ROPF 3.…”

“…Our convex relaxation approach has been further developed in a number of recent papers to handle other energyrelated optimization problems such as multi-period optimal dispatch [21], state estimation in power systems [22], optimal power flow with distributed energy storage dynamics [23], transmission system planning [24], optimal charging of plugin hybrid electric vehicles [25], distributed control for power networks [26], and optimal power balance under uncertainty [27]. Moreover, the SDP program proposed in our work [14] has been also explored by various researchers in the context of OPF to both enhance the underlying theory and resolve practical issues.…”

Modeling and analysis of stochastic AC-OPF based on SDP relaxation technique

“…A chance-constrained AC-OPF model was proposed in [18] to handle power uncertain injection, but the linearization method to obtain the sensitivity matrix may reduce the computation accuracy. A decoupled method to solve stochastic AC-OPF was developed in [15] based on the zero duality gap assumption of [12]. As the zero duality gap assumption is not always satisfied for arbitrary power systems, the practical application of this method may be restricted.…”

“…Such kind of units include large-scale coal-fired or diesel-fired generators. The re-dispatch units including ESS, tie line or gas-fired generator are of high regulation capability and flexible enough to alleviate the power imbalance caused by the uncertainty of load or wind power generation [15].…”

Optimal economic power dispatch in the presence of intermittent renewable energy sources

“…The challenges to optimal power flows are reported by Momoh et al in [5]. Examples of some methods that are used to solve the optimal power flow include, nonlinear programming [6]- [9], quadratic programming [10], Newton's method [11], [12], heuristic and swarm intelligence [13], [14], interior point method [15], [16], and linear programming (LP) based methods [17]- [20]. Amongst these methods, LP based methods are recognized as viable and promising tools in solving the constrained economic dispatch problem.…”