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Adaptive flexibility function in smart energy systems: A linearized price-demand mapping approach

Seyed Shahabaldin Tohidi, Henrik Madsen, Georgios Tsaousoglou, Tobias K. S. Ritschel

TL;DR

The paper tackles price–demand mapping in price-responsive energy systems with parametric uncertainty in the flexibility function. It develops an adaptive mechanism using a linearized price–demand mapping and a projection-based update with Lyapunov stability to force $D_t$ to follow $D_{ref_t}$. Key contributions include a deterministic linearized model with known sign of $a_t$ and $b_t$, an optimal price generator under known parameters, and a fully adaptive law with projection that guarantees bounded $u_t$ and convergence of $e_t=\mathcal{X}_t-\mathcal{Y}_t$ and $D_t$ to $D_{ref_t}$. The approach enables plug‑and‑play flexibility services for aggregators and smart buildings without requiring persistent excitation or explicit parameter identification.

Abstract

This paper proposes an adaptive mechanism for price signal generation using a piecewise linear approximation of a flexibility function with unknown parameters. In this adaptive approach, the price signal is parameterized and the parameters are changed adaptively such that the output of the flexibility function follows the reference demand signal provided by the involved aggregator. This is guaranteed using the Lyapunov stability theorem. The proposed method does not require an estimation algorithm for unknown parameters, that eliminates the need for persistency of excitation of signals, and consequently, simplifies offering the flexibility services. Furthermore, boundedness of the price signal is ensured using a projection algorithm in the adaptive system. We present simulation results that demonstrate the price generation results using the proposed approaches.

Adaptive flexibility function in smart energy systems: A linearized price-demand mapping approach

TL;DR

The paper tackles price–demand mapping in price-responsive energy systems with parametric uncertainty in the flexibility function. It develops an adaptive mechanism using a linearized price–demand mapping and a projection-based update with Lyapunov stability to force to follow . Key contributions include a deterministic linearized model with known sign of and , an optimal price generator under known parameters, and a fully adaptive law with projection that guarantees bounded and convergence of and to . The approach enables plug‑and‑play flexibility services for aggregators and smart buildings without requiring persistent excitation or explicit parameter identification.

Abstract

This paper proposes an adaptive mechanism for price signal generation using a piecewise linear approximation of a flexibility function with unknown parameters. In this adaptive approach, the price signal is parameterized and the parameters are changed adaptively such that the output of the flexibility function follows the reference demand signal provided by the involved aggregator. This is guaranteed using the Lyapunov stability theorem. The proposed method does not require an estimation algorithm for unknown parameters, that eliminates the need for persistency of excitation of signals, and consequently, simplifies offering the flexibility services. Furthermore, boundedness of the price signal is ensured using a projection algorithm in the adaptive system. We present simulation results that demonstrate the price generation results using the proposed approaches.
Paper Structure (7 sections, 3 theorems, 26 equations, 6 figures, 2 algorithms)

This paper contains 7 sections, 3 theorems, 26 equations, 6 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. FLEXIBILITY FUNCTION
  3. DETERMINISTIC LINEARIZED FLEXIBILITY FUNCTION
  4. OPTIMAL PRICE GENERATOR
  5. ADAPTIVE FLEXIBILITY FUNCTION
  6. SIMULATION RESULTS
  7. CONCLUSIONS

Key Result

Lemma 1

If $\dot{\theta}=\text{Proj}(\theta,Y)$ with initial conditions $\theta(0)\in \Omega_{\theta}=\{\theta\in \mathbb{R}|h(\theta)\leq 1\}$, where $h(\theta):\mathbb{R}\rightarrow \mathbb{R}$ is a convex function, then $\theta\in \Omega_{\theta}$ for $\forall t\geq 0$.

Figures (6)

  • Figure 1: Schematic of nonlinear functions of the flexibility function and their linear approximations.
  • Figure 2: Block diagram of the proposed adaptive flexibility function.
  • Figure 3: Price signal generation using the controller (\ref{['eq:9']}) without bounds on the price signal.
  • Figure 4: Price signal generation using the controller (\ref{['eq:9']}) with bounds on the price signal.
  • Figure 5: Price signal generation using Implementation procedure 1.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • ...and 4 more