In this chapter, we introduce a networked Cournot model for studying the impact of regula- tory objectives on the outcomes in electricity markets. In particular, the model we introduce formulates a game between the electricity market maker (or the ISO) and generators. Within this game, our main results explore the contrasts between three natural market maker ob- jectives – social welfare, residual social welfare, and consumer surplus. The results in this chapter reveal that the design of the market has significant implications on both the ex- istence and form of equilibria. In particular, equilibria might not exist when the market maker maximizes the consumer surplus and the network is capacity constrained. Further, even when equilibria exist, the equilibrium allocation of power flows can be completely dif- ferent under the three market maker objectives. Hence, the results in this chapter highlight that design of market maker objective is delicate and needs to be further investigated in a principled manner.
Chapter 6
Placing energy storage in a grid for
load-shifting
An optimization or game over a time horizon reduces to a one stage problem when the states of the system are not coupled across time. The problems considered in Chapters 2 – 5 belong to this category. In this chapter, we introduce electric energy storage that couples the system states across time. Energy storage has many potential applications in power systems. On a fast time scale (on a seconds to minutes scale), it can mitigate intermittency of renewable sources like wind and solar. On a slower time scale (across hours), it can flatten out generation profile rather than supply simply following demand. In this chapter, we concentrate on the second application, that is often referred to as load-shifting. Recall that cost of conventional generation is often quadratic and hence convex. Given such a convex cost, a flatter generation profile reduces total cost over a time horizon. Our focus is on placement and sizing of storage resources across a network to reduce the system-wide generation cost, given an available storage budget. The investment decision problem, by construction, is an infinite horizon problem. With cyclic variation in demand, it is sufficient to optimize the cost over one time period of the cycle. We do two studies in this chapter: (1) simulations using a semidefinite relaxation of AC optimal power flow on IEEE benchmark systems, (2) theoretical characterization of a property of the optimal placement using DC power flow approximations.
6.1
Background
6.1.1
Motivation
One key difference between electricity and other commodities is the concept of inventory, i.e., ability to store excess supply at one point in time and use that in conjunction with current supply to serve demand. This is precisely the flexibility that electric energy storage would provide to the power grid; in essence balancing any realized demand with instantaneous supply would no longer be necessary. This flexibility is envisioned to have many potential applications to the grid, see [172, 173] for a detailed survey. There has been much interest in building the physical devices; technologies such as pumped hydro, compressed air and Lithium ion electrochemical batteries have shown promise. No doubt, grid scale storage is still very expensive to deploy. However, their costs have shown significant drops over the last decade or so [174, 175]. For a more comprehensive literature review on storage technologies, we refer the reader to [172, 176–183]. This chapter is devoted to integration of storage in the power grid.
As argued before, storage can reduce variability of intermittent sources of energy like wind or solar [184–187]. At slower time scales, it can be used for load shifting [174, 180], i.e., generate when it is cheaper and use storage dynamics to follow the demand. Our focus in this chapter is on the latter. In this setting, there are two natural questions to ask: (a) What is the optimal investment policy for storage? Where to place them, and how to size them? (b) Once installed, what is the optimal control policy for the storage as well as the generation schedule to minimize generation costs? In this chapter, we formulate both problems for slower time-scales in a common framework and present results on the optimal placement, sizing and control of storage units.
6.1.2
Prior work in this area
Optimal control policy for installed storage units has received a lot of attention recently. While the authors in [188–190] examine the control of a single storage device without a
network, the authors in [191, 192] explicitly model the role of the networks in the operation of distributed storage resources. Storage resources at each node in the network are assumed to be known a priori in these settings.
Sizing of storage devices has been studied in the literature too. The works in [193, 194] use purely economic arguments, without explicitly considering the network constraints of the physical system. Authors in [189, 195] have looked at optimal sizing of storage devices in single-bus power system, while Kanoria et al. [191] compute the effect of sizing of distributed storage resources to optimize generation cost for specific networks.
6.1.3
Our Contribution
In this chapter, we study the investment decision problem of placement and sizing of storage in power networks. The formulation, however, builds the investment problem on top of an optimal control problem for storage. We present this formulation in Section 6.2 where the objective is to minimize system-wide general cost subject to an available storage budget. The works in [189,191] consider a similar control problem over an infinite horizon. Since aggregate demands over large geographical locations often show periodicity [196], we effectively reduce this problem to an optimization over one time period. The generators have finite capacities with convex nondecreasing costs [8, 190, 191]. We model the network, once using a conic relaxation of the AC power flow equations. Next, we use the linearized DC power flow model to simplify the formulation.
The semidefinite relaxation in Section 6.3 attempts to find some properties of the optimal placement of storage in the network. Our results here indicate that optimal storage placement in the absence of line-flow limits is largely dependent on the network structure and fairly robust to the position of renewable generation in the network. The locations (or buses) where a significant amount of storage is allocated does not change much as the total storage budget for the system is increased. However, the line-flow limits have a significant effect on where the storage is placed. When conventional generation is changed to wind generation with zero marginal cost, the distribution of storage roughly remains similar to the case of
conventional generation. In this study, we assumed perfectly efficient storage systems. The focus of the analysis with the DC power flow approximation in Section 6.4 is the derivation of a structural result on the distribution of storage. Our main contribution in this section is the result in Theorem 17: when minimizing a convex and nondecreasing generation cost with any fixed available storage budget over a slow time-scale of operation, there always exists an optimal storage allocation that assigns zero storage at nodes with only generation that connect via single transmission lines to the rest of the network. This holds for arbitrary demand profiles and other network parameters. The result provides (partial) analytic justification of the observation made empirically in Section 6.3 that optimal storage allocation seldom places storage capacities at generator-only buses.
We finally conclude in Section 6.5 with directions for future work.