Trajectory-Independent Flexibility Envelopes of Energy-Constrained Systems with State-Dependent Losses

TL;DR

The paper addresses the risk that trajectory-dependent energy envelopes fail to guarantee state-constraint satisfaction for energy-constrained loads with state-dependent losses. It introduces Trajectory-Independent (TI) energy envelopes and derives formulations for uni-dimensional and multi-dimensional systems, including distributed and centralized operation, supported by a building-thermal-dynamics case study. TI envelopes are shown to be more conservative than TD envelopes but guarantee comfort/temperature constraints across all admissible trajectories, with centralization advantageous for strongly coupled loads. Case studies on SwissHouse archetypes and a 9-room building reveal that poorly insulated or lightly constructed buildings experience the largest divergence between TI and TD envelopes, highlighting the practical need for TI envelopes in operation. The work provides a convex-optimization framework to compute TI bounds and suggests future extensions to solar gains, other energy systems, and optimal TI envelope design.

Abstract

As non-dispatchable renewable power units become prominent in electric power grids, demand-side flexibility appears as a key element of future power systems' operation. Power and energy bounds are intuitive metrics to describe the flexibility of energy-constrained loads. However, to be used in operation, any power consumption trajectory fulfilling the power and energy bounds must necessarily fulfill the load's constraints. In this paper, we demonstrate that energy bounds defined as the minimum and maximum energy consumption potential of a load with state-dependent losses are Trajectory-Dependent (TD), i.e., for any energy value in the bounds a feasible power trajectory exists, but not all power trajectories enclosed in the energy envelopes satisfy the load's constraints. To guarantee the satisfaction of load constraints for all trajectories, we define Trajectory-Independent (TI) energy bounds. We present TI envelope formulations for individual loads, as well as physically coupled loads and assess the proposed formulations in a building heating system, a system with state-dependent losses. We find that using a TD envelope as energy bounds in operation may yield room temperature up to 3.8°C higher and 3.4°C lower than admissible. Overall, poorly insulated buildings observe a TI energy envelope that differs significantly from their TD envelope.

Figures (8)

• Figure 1: On the left, an example of a TD energy flexibility envelope of a one-zone building, defined by the gray lines, with two power consumption trajectories (Scenario A and B). On the right, their resulting room temperatures, with the temperature boundaries in gray.
• Figure 2: Largest 2-dimensional box, represented in striped red, inside a polytope, represented in blue. The blue lines describe the limits of the polytope.
• Figure 3: Example of the TI and TD flexibility envelopes of a one-zone building (SwissHouse).
• Figure 4: Worst thermal discomfort for different Swiss building archetypes, when consuming as much (upper plot) and as little energy as possible (lower plot) in the TD envelope.
• Figure 5: Reduction (median) of the flexibility region between the TI and the TD envelopes for different Swiss building archetypes and prediction horizons.

Theorems & Definitions (9)

• Theorem 1
• Remark 1
• Remark 2: Connection to TD energy envelope
• Theorem 2: Coupled Energy Envelopes
• Remark 3
• Theorem 3
• Remark 4
• Theorem 4
• Remark 5