Stability analysis of nonlinear stochastic flexibility function in smart energy systems

TL;DR

The paper addresses the stability of a nonlinear stochastic Flexibility Function that maps price signals to demand and stores flexibility as a state of charge $X_t\in[0,1]$, formulating both a deterministic and an Itô stochastic model. The authors cast the model into an input-state-output form, derive equilibrium conditions via $f(X^*)=-g(u^*)$, and prove, using Lyapunov methods, that the deterministic system is asymptotically stable around its equilibria. They extend the analysis to the stochastic setting, employing Kushner-type stochastic stability theory to establish boundedness and almost-sure stability of the equilibria for small diffusion $\sigma_x$, supported by a tailored Lyapunov function and stochastic generator $\mathcal{L}$. Simulations based on an energy-zone model and real-world data demonstrate convergence to equilibria, stationary distributions, and the effect of price and baseline on storage behavior, confirming the theoretical results and illustrating practical applicability in transactive control and demand-side management.

Abstract

Demand-side management provides a great potential for improving the efficiency and reliability of energy systems. This requires a mechanism to connect the market level and the demand side. The flexibility function is a novel approach that bridges the gap between the markets and the dynamics of physical assets at the lower levels of the energy systems and activates demand-side flexibility with the purpose of decision-making as well as for offering a new framework for balancing and grid services. Employing this function as a key for many decision-making and control algorithms reveals that a mathematically rigorous stability analysis is required for it. In this paper, we investigate the stability properties of two nonlinear flexibility functions, as a dynamic mapping between electricity price and power consumption. Specifically, we analyze the stability of a deterministic flexibility function and an Itô stochastic flexibility function. Simulation results are also provided to demonstrate the dynamics of the flexibility functions and to show that the analytical results hold.

Figures (11)

• Figure 1: Schematic of nonlinear functions of the flexibility function for the stability analysis. The functions $f$ and $g$ are selected to demonstrate the analysis.
• Figure 2: Trajectories of state of charge, based on changes in initial state, price, and baseline.
• Figure 3: State trajectories of the systems in Example 1-3.
• Figure 4: State trajectories of the flexibility function with different initial conditions (top panel) and different baseline (bottom panel). state trajectories affected by $u^*=0$ and $u^*=1$ are depicted by solid lines and dashed lines, respectively.
• Figure 5: Probability density function (pdf) of the state of the flexibility function concerning time.

Theorems & Definitions (2)

• Definition 1
• Definition 2