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Cost-optimal Management of a Residential Heating System With a Geothermal Energy Storage Under Uncertainty

Paul Honore Takam, Ralf Wunderlich

TL;DR

The study tackles cost-optimal operation of a residential heating system that features a large geothermal energy storage (GES) interacting with an internal storage (IES) under uncertainty in residual demand and fuel prices. It introduces a nonstandard continuous-time model where the GES temperature dynamics are governed by a parabolic PDE with convection, requiring model order reduction to a low-dimensional ODE system before time discretization. This leads to a Markov decision process (MDP) with state-dependent action constraints, which the authors solve by discretizing the state space and applying dynamic programming to obtain approximate value functions and optimal control rules. Numerical experiments demonstrate the approach's ability to generate sensible policies and paths for various storage levels, while highlighting its potential for extension to more detailed 3D GES models and reinforcement learning-based solution methods. The practical impact lies in providing a tractable framework for design and operation of geothermal-integrated residential heating systems under uncertainty, with explicit consideration of interstorage heat transfer and temporal temperature distributions.

Abstract

In this paper, we consider a residential heating system with renewable and non-renewable heat generation and different consumption units and investigate a stochastic optimal control problem for its cost-optimal management. As a special feature, the heating system is equipped with a geothermal storage that enables the intertemporal transfer of thermal energy by storing surplus heat for later use. In addition to the numerous technical challenges, economic issues such as cost-optimal control also play a central role in the design and operation of such systems. The latter leads to challenging mathematical optimization problems, as the response of the storage to charging and discharging decisions depends on the spatial temperature distribution in the storage. We take into account uncertainties regarding randomly fluctuating heat generation from renewable energies and the environmental conditions that determine heat demand and supply. The dynamics of the multidimensional controlled state processes is governed by a partial, a random ordinary and two stochastic differential equations. We first apply a spatial discretization to the partial differential equation and use model reduction techniques to reduce the dimension of the associated system of ordinary differential equations. Finally, a time-discretization leads to a Markov decision process for which we apply a state discretization to determine approximations of the cost-optimal control and the associated value function.

Cost-optimal Management of a Residential Heating System With a Geothermal Energy Storage Under Uncertainty

TL;DR

The study tackles cost-optimal operation of a residential heating system that features a large geothermal energy storage (GES) interacting with an internal storage (IES) under uncertainty in residual demand and fuel prices. It introduces a nonstandard continuous-time model where the GES temperature dynamics are governed by a parabolic PDE with convection, requiring model order reduction to a low-dimensional ODE system before time discretization. This leads to a Markov decision process (MDP) with state-dependent action constraints, which the authors solve by discretizing the state space and applying dynamic programming to obtain approximate value functions and optimal control rules. Numerical experiments demonstrate the approach's ability to generate sensible policies and paths for various storage levels, while highlighting its potential for extension to more detailed 3D GES models and reinforcement learning-based solution methods. The practical impact lies in providing a tractable framework for design and operation of geothermal-integrated residential heating systems under uncertainty, with explicit consideration of interstorage heat transfer and temporal temperature distributions.

Abstract

In this paper, we consider a residential heating system with renewable and non-renewable heat generation and different consumption units and investigate a stochastic optimal control problem for its cost-optimal management. As a special feature, the heating system is equipped with a geothermal storage that enables the intertemporal transfer of thermal energy by storing surplus heat for later use. In addition to the numerous technical challenges, economic issues such as cost-optimal control also play a central role in the design and operation of such systems. The latter leads to challenging mathematical optimization problems, as the response of the storage to charging and discharging decisions depends on the spatial temperature distribution in the storage. We take into account uncertainties regarding randomly fluctuating heat generation from renewable energies and the environmental conditions that determine heat demand and supply. The dynamics of the multidimensional controlled state processes is governed by a partial, a random ordinary and two stochastic differential equations. We first apply a spatial discretization to the partial differential equation and use model reduction techniques to reduce the dimension of the associated system of ordinary differential equations. Finally, a time-discretization leads to a Markov decision process for which we apply a state discretization to determine approximations of the cost-optimal control and the associated value function.

Paper Structure

This paper contains 54 sections, 3 theorems, 56 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review on geothermal energy storage
  3. Literature review on optimal energy storage management
  4. Literature review on optimal management of heating systems
  5. Literature review on stochastic optimal control
  6. Our contribution
  7. Paper organization
  8. Residential Heating System
  9. Residual demand
  10. Internal energy storage
  11. Fuel-fired boiler
  12. Geothermal energy storage
  13. Control System
  14. State
  15. Control
  16. ...and 39 more sections

Key Result

Proposition 6.3

Under Assumptions Ass:ConstantControl and Ass:ConstantParameter, there exists a sequence of independent three-dimensional standard normally distributed random vectors for $(\pazocal{B}_n)_{n=1,\ldots,N}$ with $\pazocal{B}_{n}= (\pazocal{B}^R_n,\pazocal{B}^F_n,\pazocal{B}^P_n)^\prime\in\pazocal{N}(0_ for $n=0,\ldots N-1$. Here, $\pazocal{T}:\{0,\ldots,N-1\}\times \pazocal{X}\times\pazocal{K}\times

Figures (12)

  • Figure 1.1: Simplified model of a residential heating system. Sect. \ref{['sec:Heating-S']} introduces the notation and gives explanation.
  • Figure 4.1: Residual demand $\widetilde{R}$ over a period of one year (left) and a zoom on a week in mid-April (right) with parameters $\beta=0.5$, $\sigma^R=2$, $\mu_R^0=5$, $\mu_R^2=20$, $\mu_R^2=1.5$, $\delta_R^1=365$ days, and $\delta_R^2=1$ day. Blue solid line for residual demand $\widetilde{R}$, red and brown solid lines show the yearly component and the seasonality function $\mu_R$ with combined yearly and daily components. The long-term mean level $\mu_R^0$ is shown as a black solid line.
  • Figure 4.2: Two-dimensional model of the geothermal storage: decomposition of the domain $\pazocal{D}$, boundary and interface conditions. Red arrows indicate the direction of the flow.
  • Figure 4.3: Changes of thermal energy in the internal storage
  • Figure 6.1: Characterization of the set of feasible control $\pazocal{K}^Y(n,x)$ for $\ell=2$
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 5.1
  • Proposition 6.3: Transition operator
  • proof
  • Remark 6.5
  • Lemma 6.6
  • Remark 6.7
  • Theorem 6.8
  • Remark 6.9
  • proof