[Paper Review] Distributed Storage for Intermittent Energy Sources: Control Design and Performance Limits

Yashodhan Kanoria, Andrea Montanari+2 more|arXiv (Cornell University)|Oct 20, 2011

Microgrid Control and Optimization7 references4 citations

TL;DR

This paper proposes a linear-quadratic (LQ) control framework for distributed energy storage and transmission in power grids with intermittent renewable sources. It proves asymptotic optimality of LQ strategies in 1D and 2D grid topologies and quantifies performance limits as a function of storage and transmission capacity, showing that optimal performance scales with the square root of storage and transmission capacity under i.i.d. net generation fluctuations.

ABSTRACT

One of the most important challenges in the integration of renewable energy sources into the power grid lies in their `intermittent' nature. The power output of sources like wind and solar varies with time and location due to factors that cannot be controlled by the provider. Two strategies have been proposed to hedge against this variability: 1) use energy storage systems to effectively average the produced power over time; 2) exploit distributed generation to effectively average production over location. We introduce a network model to study the optimal use of storage and transmission resources in the presence of random energy sources. We propose a Linear-Quadratic based methodology to design control strategies, and we show that these strategies are asymptotically optimal for some simple network topologies. For these topologies, the dependence of optimal performance on storage and transmission capacity is explicitly quantified.

Motivation & Objective

• Address the challenge of integrating intermittent renewable energy sources—like wind and solar—into power grids due to their time- and location-dependent variability.
• Investigate the joint impact of distributed energy storage and transmission capacity on system performance in averaging out renewable generation variability.
• Design control strategies that optimally balance local storage and networked power exchange to minimize imbalance in energy supply and demand.
• Quantify the fundamental performance limits of such systems in terms of storage and transmission capacity, particularly in asymptotic regimes.
• Establish theoretical optimality of LQ-based control strategies in simple network topologies like 1D and 2D grids.

Proposed method

• Model the power grid as a weighted graph with nodes representing buses and edges representing transmission lines, using discrete time slots for dynamic analysis.
• Define net generation at each node as the difference between renewable production and local demand, modeled as a stationary stochastic process.
• Introduce distributed storage at each node with finite capacity, assuming no losses in energy recovery.
• Formulate a linear-quadratic (LQ) optimal control problem to minimize the long-term variance of net imbalance across the network.
• Derive control laws that determine optimal energy injection/withdrawal from storage and power exchange over transmission lines at each time step.
• Use probabilistic tools, including sub-Gaussian tail bounds and exponential moment methods, to analyze performance and derive lower and upper bounds on the probability of large imbalances.

Experimental results

Research questions

• RQ1 How does the combination of distributed storage and transmission capacity affect the system's ability to smooth out intermittent renewable generation?
• RQ2 What is the fundamental performance limit of such systems in terms of minimizing imbalance variance, and how does it scale with storage and transmission capacity?
• RQ3 Can linear-quadratic (LQ) control strategies achieve asymptotic optimality in simple network topologies like 1D and 2D grids?
• RQ4 What is the role of spatial correlation and geographical distribution in reducing variability through network averaging?
• RQ5 How tight are the theoretical performance bounds, especially in the asymptotic regime of large networks?

Key findings

• The LQ-based control strategy is asymptotically optimal for 1D and 2D grid topologies under i.i.d. net generation processes.
• Performance limits are quantified: the probability of large imbalance decays exponentially with the system size, with the exponent depending on the ratio of transmission capacity to storage capacity.
• For a 1D grid, the optimal performance scales as $ rac{1}{ ext{variance}} o rac{C}{ ext{variance}} $, where $ C $ is the transmission capacity, and the decay rate is $ rac{C}{ ext{variance}} $, indicating that performance improves with higher transmission capacity.
• The upper bound on the probability of imbalance exceeding a threshold is $ rac{1}{4} imes ext{exp}igrace -rac{ ext{const} imes C imes ext{max}(C,S)}{ ext{variance}} igrace $, showing exponential decay in the product of transmission and storage capacity.
• The lower bound on the same probability matches the upper bound up to constants in the exponent, proving that the bound is tight in the exponential order.
• The performance limit depends on the square root of the storage and transmission capacities, indicating diminishing returns beyond a certain scale.