Convex Optimization of Power Systems

The day-to-day and long-term operation of power systems is made up of a sprawling collection of engineering and economic tasks that subsumes most of the contents of this book. This chapter does not address the full scope of power system operation but rather a handful of relevant problems that are amenable to convex optimization. As will be seen, many of these problems consist of details layered on top of the optimal power flow formulations of Chapter 3. The interested reader is referred to Wood and Wollenberg [1] for broad coverage of power system operation.

Of particular relevance to this chapter is Theorem 3.1 of Section 3.3.1, which guarantees that under certain conditions, the SOC optimal power flow relaxation is exact in radial networks. It is therefore a great boon that distribution systems, the portions of power systems that convey low-voltage electricity from substations to end users, are almost always operated radially. Until recently, distribution systems were essentially passive, predictable energy consumers. The shift from fuel-based centralized power plants to distributed and renewable generation and the new, active role of loads like electric vehicles and smart buildings are transforming distribution systems into highly actuated, potentially volatile consumers and producers of electric power. Consequently, many of the formulations in this chapter are both highly tractable and relevant when specialized to radial networks. This is especially true in Section 4.4, which deals exclusively with power flow in radial networks.

Multi-period optimal power flow

Optimal power flow routines are run every few minutes to update device and resource settings in response to the constantly changing conditions of power systems. The dynamic couplings present over these time scales justify linking these routines over successive time periods. This section formulates the multi-period optimal power flow and elucidates its application through generator ramp constraints and energy storage.

Essentially, multi-period optimal power flow is just a sequence of ordinary optimal power flow routines strung together by dynamic costs and constraints.