Model order reduction for the input–output behavior of a geothermal energy storage

This article investigates the input–output behavior of geothermal storage systems which describes the response of the storage to charging and discharging operations. Geothermal storage tanks represent an important class of thermal storage systems and enable extremely efficient operation of heating and cooling systems in individual buildings as well as in district heating systems. They allow heat or cold to be stored in such a way that it can be used hours, days, weeks, or even months later. This helps to shift heat supply and demand over time and to mitigate temporal fluctuations, and is beneficial for space heating, domestic or industrial water heating, and electricity generation. Thermal storage systems can significantly increase both the flexibility and performance of district heating systems and improve the integration of intermittent renewable energy sources into heating networks (see Guelpa and Verda [1], Kitapbayev et al. [2]). They also support peak shaving in power grids, i.e., mitigating peaks by converting electrical energy into thermal energy (power-to-heat). Several geothermal reservoirs can be pooled to form a virtual power plant that provides the necessary capacity for participation in the balancing energy market.

We consider geothermal storage tanks as depicted in Fig. 1 and as presented in detail in our previous work [3, 4]. Such storage tanks are increasingly important and are quite attractive for heating systems in residential buildings, as construction and maintenance are relatively inexpensive. Moreover, they can be integrated into both new and existing buildings. To construct a geothermal storage system, a storage tank with a defined volume is filled with soil and insulated against the surrounding soil. Thermal energy is stored by raising the temperature of the soil inside the storage tank. It is charged and discharged through pipe heat exchangers (PHX) filled with a fluid like water. These PHXs are connected to a heating system in the building, especially to water tanks used for storing heat, or directly to a solar collector. The liquid that transports the thermal energy is moved by pumps. Some of them are heat pumps that raise the fluid temperature to a higher level by converting electrical energy into thermal energy.

A key feature of the geothermal reservoir under consideration is that it is not insulated at the bottom, allowing thermal energy to flow to deeper layers, as shown in Fig. 2. This allows for expansion of the storage capacity, as this heat can be recovered if the storage tank is sufficiently discharged (cooled) and a heat flow is induced back to the storage tank. The inevitable losses due to diffusion into the environment can be compensated, as such geothermal storage units benefit from the higher temperatures in deeper soil layers and can therefore also serve as production units similar to borehole heat exchangers. This is particularly useful in colder climates, such as in many parts of Europe or North America, where soil temperatures at a depth of only 10 m remain relatively stable around \(10 \,{^{\circ }\text {C}}\) for most of the year.

For efficient operation of geothermal storage systems within residential systems, it is necessary to know certain aggregated characteristics of the spatial temperature distribution in the storage tank, their dynamics, and their responses to charging and discharging decisions. It must be determined whether to pump heated or cooled fluid through the PHXs to the storage tank or whether the pumps should be switched off. Furthermore, an appropriate fluid temperature needs to be set during the operation of the pumps. An example of such an aggregated characteristic is the spatially averaged temperature in the storage medium from which one can derive the amount of available thermal energy that can be stored in or extracted from the storage. Another example is the average temperature at the outlet, which allows to determine the amount of energy injected into or withdrawn from the storage. Furthermore, the average temperature at the bottom of the storage allows to quantify the heat transfer to and from the ground via the open bottom boundary. For more details, we refer to Sect. 4.

The above aggregated characteristics can be computed by post-processing the spatio-temporal temperature distribution in the storage. The latter can be obtained by solving the governing linear heat equation with convection and appropriate boundary and interface conditions. This is explained in Sects. 2 and 3. In this study, we utilize a stylized mathematical model of the storage that captures the physical properties needed to describe its input–output behavior, but does not consider all the engineering details. We use finite difference methods to semidiscretize the partial differential equation (PDE) w.r.t. the spatial variables. This approach is also known as ’method of lines’ and leads to a high-dimensional system of ODEs. We refer to [3, 4] for details and a stability analysis of the finite difference scheme as well as results of extensive numerical experiments. Further, we refer to a previous study in Bähr et al. [5, 6], where the storage was not considered isolated but embedded in the surrounding domain and the interaction between geothermal storage and the environment is examined. In that study, the focus was on the numerical simulation of the long-term behavior of the spatial temperature distribution. To simplify the analysis, a simple source term was used to describe charging and discharging, instead of PHXs. Here, we focus on the short-term behavior of the spatial temperature distribution, in particular its response to charging and discharging processes. We choose the computational domain to be the storage depicted by a solid black rectangle in Fig. 2. In our stylized model, we do not consider the surrounding medium but set appropriate boundary conditions to mimic the interaction between storage and environment. In contrast to [5, 6], we model PHXs which are used to charge and discharge the storage.

Literature review on modeling and simulation of geothermal storage systems  Thermal energy storage technologies can be divided into three categories which are sensible heat, latent heat, and thermo-chemical heat storage. In a sensible storage, the temperature of some medium is either increased or decreased. Latent heat storage uses a phase transition of phase-change material. Heat can be added or removed at these transitions without changing the temperature of the material. For an overview, we refer to Zayed et al. [7]. Thermo-chemical heat storage utilizes reversible chemical reactions with thermo-chemical materials.

The geothermal storage system investigated in this article is a sensible storage. It is a relatively new and specialized technology that has only been developed and deployed in the last 15 years. To the best of our knowledge, there are only a few references such as [3‐6] about mathematical modeling and numerical simulation of this type of storage tanks which we already cited above. However, the heat transfer and exchange processes between the heat exchanger and the surrounding soil do not only play a crucial role in this work, but also for the performance of ground source heat pumps in general. The latter are used to extract heat from the ground, collect it, and feed it into heating systems, whereby a storage function is not provided or is only of secondary importance. The systems are divided into horizontal and vertical ground heat exchangers for which a considerable body of engineering literature about modeling and simulation is available. A comprehensive survey is given in Zayed et al. [8].

For horizontal ground heat exchangers, Dasare et al. [9], Gao et al. [10], Hu et. al. [11], Selamat et al. [12], and Shi et al. [13] study the performance of different heat exchanger arrangements (linear-loop, spiral-coil, slinky-coil), and the effect of the velocity of the heat carrier fluid, operation modes, subsurface water flow, soil thermal conductivity, installation depth, and certain atmospheric conditions above the surface. Vertical or borehole heat exchangers are studied in Frei et al. [14], Tilley et al. [15]. Here, the authors consider analytical methods for the computation of the thermal resistance of such systems. Tilley at al. [16] consider a borehole heat exchanger combined with aquifer or water channels that improve the heat transfer rates of the storage system. The focus is on fluid transport through a dynamic porous calcite medium in which a reaction occurs between the solid matrix and the fluid. The authors use a two-dimensional model that has some properties in common with the model presented in Sect. 2.1, see also Remark 2.2, and apply analytical methods, in particular homogenization techniques. Kim et al. [17] and Brunetti et al. [18] focus on the numerical simulation.

Model order reduction & optimization of heating systems  The results of this article are an important prerequisite and indispensable preliminary work to solve optimization problems arising in the cost-optimal management of residential heating systems equipped with a geothermal storage. These are decision-making problems under uncertainty and treated mathematically in terms of stochastic optimal control problems as in Takam [3]. The goal was to find optimal decision rules that determine feedback controls, including storage’s charging and discharging operations, which minimize the expected aggregate costs from the operation of the heating system. We include stochastics in the optimization problem because uncertainties regarding the future local production of thermal energy from renewable sources, e.g., by a solar collector must be taken into account for optimal operation. Predictions of such weather-dependent variables are difficult and fraught with uncertainties. Further, the heating demand is driven by the temperature of the environment, which is also weather dependent and not exactly predictable. Another source of uncertainties is the future price of fuel such as gas and oil needed for heat production in fuel-fired boilers. Larger consumers often do not have fixed tariffs, but are affected by fluctuations of the energy prices on energy markets. The market’s inherent uncertainties market must then be taken into account in the decision-making process for the optimal operation of the system.

Working directly with the PDE or the large-scale system of ODEs for the mathematical description of the dynamics of the controlled state is prohibitively expensive, when we need to solve the stochastic optimal control problem numerically. This is our main motivation for investigating model order reduction (MOR) techniques and explained in more detail in the next paragraph. Note that this differs from deterministic control problems, and also from the possibility of simply simulating the state dynamics numerically. For the two-dimensional model under consideration, such a simulation is not too expensive from the point of view of computational time. Even for refined and three-dimensional models, this is feasible with today’s computer capabilities. Of course, MOR is also helpful there. For example, Kim et al. [17] use modal decomposition for MOR to improve computational efficiency of the simulations.

Model order reduction & stochastic optimal control  In continuous time and using the system of ODEs, the control problem’s state dynamics will be governed by a system of SDEs and ODEs. The control problem can be tackled using the established theory of controlled diffusion processes. Applying dynamic programming techniques, the desired optimal decision rules can be derived from necessary optimality conditions, which are given in terms of a highly nonlinear PDE known as Hamilton–Jacobi–Bellman equation. Here, the number of “spatial variables” is given by the number of states in the control problem which is very high. Typically, closed-form solutions are not available and one has to rely on numerical methods to solve that nonlinear PDE. This approach already suffers from the curse of dimensionality even when the number of states is small, say if it exceeds three. A typical remedy is to perform a suitable time discretization and to apply the theory of Markov decision processes as in [3] and the references therein. Then, modern machine learning techniques can help to overcome the curse of dimensionality if the number of states is moderate but not too high.

In summary, it is necessary to describe the input–output behavior of the storage by a suitable low-dimensional system of ODEs with a sufficiently high approximation accuracy. This leads to a problem of MOR, which is the focus of the present study.

Analogous model In our model in Sect. 2, we assume a piecewise constant velocity for the fluid flow in the PHX. This is often observed in real-world systems, which operate at constant velocity during charging and discharging if pumps are in operation, while the velocity is zero if pumps are switched off. Then, the high-dimensional system of ODEs constitutes a system of linear nonautonomous ODEs since the system matrix depends on time via the fluid velocity. The latter varies over time but it is piecewise constant. Thus, the obtained linear system is not linear time invariant (LTI). However, the latter is a crucial assumption for many MOR methods. In Sect. 5, we circumvent this problem by approximating the model for the geothermal storage by an analogous model, which is LTI. The key idea for the construction of such an analog is to mimic the original model by an LTI system, where pumps are always operating, so that the fluid velocity is constant at all times. During the waiting periods, we use the same type of boundary conditions at the inlet and outlet boundary as during charging and discharging. However, we choose the inlet temperature to be equal to the spatially averaged temperature in the PHX. Numerical results presented in Subsect. 7.2 show that the analogous system approximates the original system well.

Literature review on model order reduction For the derived linear LTI system, we applied MOR methods in Sect. 6. Such methods can be roughly divided into projection and truncation methods based on singular value decomposition (SVD), and moment matching methods as the Krylov subspace method. Depending on the structure of the problem, we can further subdivide SVD into two classes. The first class contains methods that are also suitable for nonlinear systems such as proper orthogonal projections (POD). The second class consists of Gramian-based approximations such as balanced truncation and Hankel approximation methods, which are suitable for linear systems. See Antoulas et al. [19, 20] and Schilders et al. [21] for a general overview.

Here, we focus on the Lyapunov balanced truncation MOR method, which is well suited for our purposes. It was first introduced by Mullis and Roberts [22] and later in the linear dynamical systems and control literature by Moore [23]. The idea of this method is firstly transform the system into an appropriate coordinate system for the state space, in which the states that are difficult to reach, that is, they require a large input energy to be reached. These are simultaneously difficult to observe, i.e., generate a small observation output energy. Then, the reduced model is obtained by truncating the states which are simultaneously difficult to reach and to observe. Among the various MOR methods, balanced truncation is characterized by the preservation of several system properties such as stability and passivity, see Pernebo and Silverman [24]. Moreover, it provides error bounds that permit an appropriate choice of the dimension of the reduced-order model depending on the desired accuracy of the approximation, see Enns [25] and Glover [26].

Besides the Lyapunov balancing method, other types of balancing techniques exist, e.g., stochastic balancing, bounded real balancing, positive real balancing, and frequency weighted balancing, see Antoulas [19] and Gugercin and Antoulas [27]. Gosea et al. [28] consider balanced truncation for linear switched systems. Benner et al. [29] present an efficient implementation of model reduction methods such as modal truncation, balanced truncation, and other balancing-related truncation techniques. Furthermore, the authors discussed various aspects of balancing-related techniques for large-scale systems, structured systems, and descriptor systems. The results presented in [29] also cover MOR techniques for time-varying as well as MOR for second- and higher-order systems. In addition, surveys on system approximation and MOR can be found in [19, 27, 30‐37] and the references therein.

Results of extensive numerical experiments justify the replacement of the original system by the LTI analogous system and show the efficiency of the applied MOR method. Only a few suitably chosen ODEs are sufficient to produce good approximations of the input–output behavior of the storage. This allows to work with reduced-order models in optimal control problems for the cost-optimal management of geothermal storage systems embedded in residential heating systems, which we investigated in [3].

Paper organization  In Sect. 2, we describe the mathematical modeling of the geothermal storage. We present a heat equation with a convection term and appropriate boundary and interface conditions that govern the dynamics of the spatial temperature distribution in the storage. Section 3 is devoted to the finite difference semidiscretization of the heat equation. In Sect. 4, we introduce aggregated characteristics of the spatio-temporal temperature distribution. Section 5 derives the approximate LTI analogous model of the geothermal storage. In Sect. 6, we start with the formulation of the general model reduction problem. Then, we present the Lyapunov balanced truncation method. Section 7 presents results of various numerical experiments, where the aggregated characteristics of the temperature distribution in storage for the original model are compared with the approximations obtained from reduced-order models. Section 8 contains our concluding remarks.

The setting is based on [4, Sect. 2]. For self-containedness and the convenience of the reader, we recall the model description in this section. The dynamics of the spatial temperature distribution in a geothermal storage can be described mathematically by a linear heat equation with a convection term and appropriate boundary and interface conditions.

We now sketch the discretization of the heat equation (1) together with the boundary and interface conditions given in (2) through (5). For details, we refer to [4, Sect. 3]. We confine ourselves to a semidiscretization in space and approximate only spatial derivatives by their respective finite differences. This approach is also known as ‘method of lines’ and leads to a high-dimensional system of ODEs for the temperatures at the grid points. The latter will serve as starting point for the model reduction in Sect. 6. For the full discretization in which time is also discretized, we refer to [4, Sect. 4], where we derive an implicit finite difference scheme and study its stability.

In many applications, it is not necessary to know the complete information about the spatio-temporal temperature distribution in the geothermal storage which can be computed using the numerical methods described in Sect. 3. This is the case in [3], which studies the cost-optimal management and control of a storage which is embedded into a residential heating system. Here, it is sufficient to know only a few aggregated characteristics of the temperature distribution, which can be computed via post-processing as in [4, Sect. 5]. For self-containedness and the convenience of the reader, we sketch some of these aggregated characteristics and describe their approximate computation based on the solution vector Y of the finite difference scheme. Below in Sect. 6, these quantities will serve as output variables of a linear dynamical system and we apply MOR techniques to construct approximations based on a small numbers of suitable chosen ODEs.

Equation (6) represents a system of n linear nonautonomous ODEs. Since some of the entries in the matrices \({A}\) and \({B}\), resulting from the discretization of convection terms in the heat equation (1), depend on the velocity \(v_0(t)\), it follows that \({A}\) and \({B}\) are time-dependent. Thus, (6) does not constitute a linear time-invariant (LTI) system. The latter is a crucial assumption for many MOR methods such as the Lyapunov balanced truncation technique that is considered below in Sect. 6. We circumvent this problem by replacing the model for the geothermal storage by a so-called analogous model which is LTI.

The key idea for the construction of such an analog is based on the observation that the assumption to our “original model” are such that it is already piecewise LTI. Recall that the fluid velocity is constant \({\overline{v}}_0\) during (dis)charging when the pump is on, and zero during waiting when the pump is off. This leads to the following two-step approximation of the original by an analogous model.

Approximation Step 1  For the analogous model, we assume that contrary to the original model, the fluid is also moving with constant velocity \({\overline{v}}_0\) during pump-off periods. During these waiting periods, in the original model, the fluid is at rest and only subject to the diffusive propagation of heat. In order to mimic this behavior of the resting fluid by a moving fluid, we assume that the temperature \(Q^{I}\) at the PHX inlet is equal to the average temperature of the fluid in the PHX \({{\overline{Q}}} ^F\). Thus, we will preserve the average temperature of the fluid, but a potential temperature gradient along the PHX is not preserved and replaced by an almost flat temperature distribution. It can be expected that the error induced by this “mixing” of the fluid temperature in the PHX is small since in real-world systems the (dis)charging periods are typically quite long. This leads to a saturation with an almost constant temperature profile along the PHX.

The initial boundary value problem for the heat equation (1) now comes with a modified boundary condition at the inlet. During waiting the homogeneous Neumann boundary condition in (3) is replaced by a nonlocal coupling condition such that the inlet boundary condition reads asThis is a “nonlocal” condition since the inlet temperature is not only specified by the local temperature distribution at the inlet boundary \(\partial {\mathcal {D}}^{I}\), but depends on the whole spatial temperature distribution in the fluid domain \({\mathcal {D}}^F\). Semidiscretization of the above boundary condition using approximation (11) of the average fluid temperature \({{\overline{Q}}} ^F\) leads to a modified input term g(t) of the system of ODEs (6). Instead of (9), it is now given byFurther, the nonzero entries \(B_{l1}\) of the input matrix B given in (8) are modified. They are now no longer time-dependent but given by the constant \( B_{l1}=\frac{a^F}{h^2_x}+\frac{{\overline{v}}_0}{h_x} \), which was already used during pump-on periods.

Approximation Step 2  In view of (12), the modified input term g depends on the state vector Y when pumping via \(C^{F}\,Y\) and can no longer be considered exogenous. Formally, the term \(C^{F}\,Y\) has to be included in \({A}Y\), which would lead to an additional contribution to the system matrix \({A}\) given by \({B}_{\bullet 1}C^{F}\) where \({B}_{\bullet 1}\) denotes the first column of \({B}\). Then, the system matrix again would be time-dependent and the system not LTI. In order to facilitate the application of MOR methods, we perform a second approximation step and formally treat \({{\overline{Q}}} ^F\) as an exogenously given quantity (such as \(Q^{I}\) and \(Q^{G}\)). This leads to a tractable approach since the Lyapunov balanced truncation technique applied in the next section generates low-dimensional systems depending only on the system matrix \({A}\) and the input matrix \({B}\), but not on the input term g, which now contains \({{\overline{Q}}} ^F\). This approach is also tractable for numerical schemes for simulating the system and from an algorithmic or implementation point of view, since given the solution Y of (6) at time t, the average fluid temperature \({{\overline{Q}}} ^F(t)\) can be computed as a linear combination of the entries of Y(t). The only requirement is to work with explicit rather than implicit time discretization of \({{\overline{Q}}} ^F\).

In Subsect. 7.2, we present results of numerical experiments indicating that apart from some small approximation errors in the PHX during waiting periods, in particular at the outlet, the other deviations are negligible.

In this section, we present results of numerical experiments on MOR for the system of ODEs (6) resulting from semidiscretization of the heat equation (1), which models the spatio-temporal temperature distribution of a geothermal storage. For describing the input–output behavior of that storage, we use the aggregated characteristics of the spatial temperature distribution introduced in Sect. 4. Further, we work with the approximation of (6) by an analogous system that is LTI as explained in Sect. 5. Recall that here it is assumed that the pump is always on and during the waiting periods the inlet temperature \(Q^{I}\) is set to be the average temperature \({{\overline{Q}}} ^F\) in the PHX fluid.

First, we perform a numerical experiment in which we investigate the accuracy of the approximation of aggregated characteristics of the original system by those from the LTI analogous system. Then, we present approximations of the aggregated characteristics obtained from reduced-order models together with error estimates provided in Theorem 6.3 for systems with zero initial conditions. Here, we apply the approach sketched at the end of Subsect. 6.2, where in a pre-processing step, we first shift the temperature scale by the constant initial temperature \(Q_0\) to obtain a linear system (13) with zero initial value for which we can compute the error bounds. Then, in a post-processing step, the inverse scale shift is applied. The experiments are based on Algorithm 6.2 and are performed for the cases of one and three PHXs, see Fig. 5. The model with three PHXs is more realistic and shows more structure of the spatial temperature distribution in the storage, whereas for one PHX that distribution is less heterogeneous. The Lyapunov equations (15) for the Gramians are solved numerically using the Matlab package mess-lyap provided by Saak et al. [42].

After explaining the experimental settings in Subsect. 7.1, we compare in Subsect. 7.2 aggregated characteristics of the original and the LTI analogous system. Then, we perform in Subsect. 7.3 an experiment in which the system output consists of two output variables, which are the spatially averaged temperatures \({{\overline{Q}}} ^M\) and \({{\overline{Q}}} ^F\) in the storage medium and in the PHX, respectively. Finally, we add a third output variable, which is in Subsect. 7.4 the spatially averaged temperature at the outlet \({{\overline{Q}}} ^{O}\). The outlet temperature \({{\overline{Q}}} ^{O}\) is interesting for the management of heating systems equipped with a geothermal storage.

Note that [3, Subsects. 4.3.1, 4.3.4, 4.3.5] contains additional results for models with one, three, and four output variables, respectively. There, the spatially averaged temperature at the bottom boundary \({{\overline{Q}}} ^B\) is taken as a fourth output variable. The latter allows to quantify the transfer of thermal energy to the environment at the bottom boundary of the geothermal storage.

In this study, we have considered the approximate description of the input–output behavior of a geothermal storage by a low-dimensional system of linear ODEs. The starting point was the mathematical modeling of the spatio-temporal temperature distribution in a two-dimensional cross-section of the storage by a linear heat equation with a convection term. By semidiscretization of that PDE w.r.t. spatial variables, we obtained a high-dimensional system of nonautonomous ODEs. The latter was approximated by an analogous LTI system. Reduced-order models in which the state dynamics are described by a low-dimensional system of linear ODEs were derived by the Lyapunov balanced truncation method. In our numerical experiments, we considered aggregated characteristics describing the input–output behavior of the storage, which are required for the operation of the geothermal storage within a residential heating system. The results show that it is possible to obtain relatively accurate approximations from reduced-order systems with only a few state variables. This allows treating the cost-optimal management of residential heating systems as a decision-making problem under uncertainty, which mathematically can be formulated as a stochastic optimal control problem. First results can be found in [3] and further findings will be published in forthcoming articles. After conducting the proof of concept of the proposed MOR method in this study, we aim to apply it to refined models based on three-dimensional models of the geothermal storage, a more detailed PHX topology and more realistic fluid-to-medium heat transfer models.