Decentralized Coordination of Energy Consumption for Smart Neighborhoods

Abstract

While the resource optimization of a single smart home is important, the aggregated effect of multiple smart homes is important to the performance of the overall system. In this chapter, we focus on the middle level of the hierarchical architecture: a set of smart homes connected to the same bus forming into a smart neighborhood. In this aggregation level, multiple smart homes are under different ownerships, and a neighborhood energy manager is responsible for coordinating their operations. Specifically, we investigate how to minimize the total energy cost of multiple smart homes in a neighborhood sharing a load serving entity. Specifically, each home has renewable generation, energy storage as well as inelastic and elastic energy loads, and the load serving entity attempts to coordinate the energy consumption of these homes in order to minimize the total energy cost within this neighborhood. We develop an online control algorithm, called LCMA, which jointly considers the energy management and demand response decisions. LCMA only needs to keep track of the current values of the underlying stochastic processes without requiring any knowledge of their statistics. Moreover, a distributed algorithm to implement LCMA is also developed, which can preserve the privacy of individual home owners. Numerical results based on real-world trace data show that our control algorithm can indeed reduce the total energy cost in the neighborhood.

Appendix

3.1.1 The Worst-case Delay for Buffered Elastic Loads

Here we prove Lemma 3.1.

Proof

For all households i, consider any slot t for which d _i, 2(t) > 0. We will show that this energy load d _i, 2(t) is served on or before time slot t + δ _i ^max by contradiction. Suppose not, then during slots τ ∈ {t + 1, …, t + δ _i ^max}, it must be that Q _i (τ) > 0. Otherwise, the energy load d _i, 2(t) would have been served before τ. Therefore, 1 _{Q(τ) > 0} = 1, and from the update Eq. (3.25) of Z _i (t), we have for all τ = {t + 1, …, t + δ _max }:

$$\displaystyle{Z_{i}(\tau +1) \geq Z_{i}(\tau ) - y_{i}(\tau ) +\varepsilon _{i}.}$$

Summing the above over τ = {t + 1, …, t + δ _i ^max} yields:

$$\displaystyle{Z_{i}(t +\delta _{ i}^{max} + 1) - Z_{ i}(t + 1) \geq -\sum _{\tau =t+1}^{t+\delta _{i}^{max} }y_{i}(\tau ) +\delta _{ i}^{max}\varepsilon _{ i}.}$$

Rearranging the terms and using the facts that Z _i (t + 1) ≥ 0 and Z _i (t + δ _i ^max + 1) ≤ Z _i ^max yields:

$$\displaystyle{ \sum _{\tau =t+1}^{t+\delta _{i}^{max} }y_{i}(\tau ) \geq \delta _{i}^{max}\varepsilon _{ i} - Z_{i}^{max}. }$$

(3.41)

Since the energy loads d _i, 2(t) are queued in a FIFO manner and Q _i (t + 1) ≤ Q _i ^max, it would be served on or before time t + δ _i ^max whenever there are at leat Q _i ^max units of energy served during τ ∈ {t + 1, …, t + δ _i ^max}. Since we have assumed that the energy loads d _i, 2(t) are not served by time t + δ _i ^max, it must be that $\sum _{\tau =t+1}^{t+\delta _{i}^{max} }y_{i}(\tau )$ < Q _i ^max. Comparing this inequality with (3.41) yields:

$$\displaystyle{Q_{i}^{max}>\delta _{ i}^{max}\varepsilon _{ i} - Z_{i}^{max},}$$

which implies that δ _i ^max < (Q _i ^max + Z _i ^max)∕ɛ _i , contradicting the definition of δ _i ^max in (3.26).