Power of Ensemble Diversity and Randomization for Energy Aggregation

Abstract

We study an ensemble of diverse (inhomogeneous) thermostatically controlled loads aggregated to provide the demand response (DR) services in a district-level energy system. Each load in the ensemble is assumed to be equipped with a random number generator switching heating/cooling on or off with a Poisson rate, r , when the load leaves the comfort zone. Ensemble diversity is modeled through inhomogeneity/disorder in the deterministic dynamics of loads. Approached from the standpoint of statistical physics, the ensemble represents a non-equilibrium system driven away from its natural steady state by the DR. The ability of the ensemble to recover by mixing faster to the steady state after its DR’s use is advantageous. The trade-off between the level of the aggregator’s control, commanding the devices to lower the rate r , and the phase-space-oscillatory deterministic dynamics is analyzed. Then, we study the effect of the load diversity, investigating four different disorder probability distributions (DPDs) ranging from the case of the Gaussian DPD to the case of the uniform with finite support DPD. We show that stronger regularity of the DPD results in faster mixing, which is similar to the Landau damping in plasma physics. Our theoretical analysis is supported by extensive numerical validation.

Mean-field control for efficient mixing of energy loads

“…Within the model considered and assuming broadcast (one way) communication from the aggregator to individual devices of their ensemble-common, adjustable switch on/off rate, we continue analysis of previous work [5,21] and investigate the power of randomization on the overall performance of the ensemble. The new significant emphasis of this manuscript is on the possibility for the Poisson rate, uniform across the ensemble for switching on/off when a device is outside the comfort zone, to be nonlinear and moreover to depend on the independently measured, instantaneous, cumulative (thus the term mean field) consumption of the ensemble.…”

“…Fourth, all solutions of Eqs. (21,22) result in non-negative λ k;± , where λ 0,+ = 0 is excluded from the perturbation series because it corresponds to the stationary state described in Section IV A. Finally, fifth, at s = 0 one recovers in Eq.…”

“…As shown in several studies [1,2,5,15,21], randomization of the control policy [1,2,5] and diversification/inhomogeneity of load types within the ensemble [15,21] both help to accelerate the mixing. In this manuscript we continue to focus on the same task (acceleration); however, we explore a new option related to the so-called Mean-Field Control [12,13,19,23].…”

“…Note that at s > 0, the correction (to the bare solution) is always real and positive, thus leading to a monotonic increase of the relaxation rate, Re(λ (21,22).…”

“…which accounts for all of the "±"-modes, contrary to N ↑ (t), and is dependent only on the "−"-branch of the spectrum Eq. (21). (Coefficient A in Eq.…”