An optimal home energy management system for modulating heat pumps and photovoltaic systems
TThis work is licensed under a Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” license.
An optimal home energy management system for modulating heatpumps and photovoltaic systems
Lissy Langer a, ∗ , Thomas Volling a a Work Group Production and Operations Management (POM), Technische Universit¨at Berlin, Straße des 17. Juni 135,10623 Berlin, Germany
Abstract E ffi cient residential sector coupling plays a key role in supporting the energy transition. In thisstudy, we analyze the structural properties associated with the optimal control of a home energymanagement system and the e ff ects of common technological configurations and objectives. Weconduct this study by modeling a representative building with a modulating air-sourced heatpump, a photovoltaic (PV) system, a battery, and thermal storage systems for floor heating andhot-water supply. In addition, we allow grid feed-in by assuming fixed feed-in tari ff s and consideruser comfort. In our numerical analysis, we find that the battery, naturally, is the essential build-ing block for improving self-su ffi ciency. However, in order to use the PV surplus e ffi ciently gridfeed-in is necessary. The commonly considered objective of maximizing self-consumption is noteconomically viable under the given tari ff structure; however, close-to-optimal performance andsignificant reduction in solution times can be achieved by maximizing self-su ffi ciency. Based onoptimal control and considering seasonal e ff ects, the dominant order of PV distribution and thetarget states of charge of the storage systems can be derived. Using a rolling horizon approach,the solution time can be reduced to less than 1 min (achieving a time resolution of 1 h per year).By evaluating the value of information, we find that the common value of 24 h for the predic-tion and control horizons results in unintended but avoidable end-of-horizon e ff ects. Our inputdata and mixed-integer linear model developed using the Julia JuMP programming language areavailable in an open-source manner. Keywords:
Heat pump, Photovoltaics (PV), Demand-side flexibility, Thermal energy storage,Model predictive control (MPC), Mixed-integer linear programming (MILP)
1. Introduction
The German federal government’s
Climate Action Plan 2050 describes the essential stepsrequired for decarbonizing the energy and building sector in Germany [14]. The target for the ∗ [email protected] Preprint submitted to Applied Energy September 15, 2020 a r X i v : . [ ee ss . S Y ] S e p uilding sector is to develop a virtually climate-neutral housing stock by 2050. Because of thelongevity of buildings, emission reductions of approximately 66% are necessary by 2030 whencompared with the emissions in 1990. Hence, from 2020, low-interest loans and investmentsubsidies of up to 45% will be o ff ered to the homeowners investing in renewable heating [12].Thus, the goal is to improve the energy e ffi ciency and the share of renewable energy sourceswith respect to the total energy consumption [14]. In 2019, the photovoltaic (PV) capacity inGermany increased by 8% and became almost 50 GWp [13]. Approximately 76% of the PVsystems are installed in the residential sector and are smaller than 10 kWp [5]. Most of theenergy consumption in the residential sector can be attributed to the thermal demand and notelectricity demand (69% for heating and 15% for hot-water supply) [33]. Therefore, e ffi cientconversion technologies that exploit the full potential of residential PV systems are needed [26].Heat pumps, especially when combined with thermal storage systems, are considered to be keyfacilitators because they provide an e ffi cient power-to-heat ratio [12, 17]. In 2018, 44% of allthe newly approved residential buildings in Germany were equipped with a heat pump (82% ofwhich were air-sourced) [34, 6]. ¨Oko-Institut and Agora Energiewende predict that 3–5 millionheat pumps will be installed in Germany alone by 2030 [1, 3].A major challenge associated with the usage of renewable energy sources is their highlyuncertain and intermittent nature. The fluctuations in supply can be mitigated by demand-sideflexibility; this is typically achieved in the residential sector by coupling electricity and heat [26,27, 31]. Thus, the thermal and battery storage systems are combined with heat pumps operatingin a flexible manner with varying intensities (modulating). In Germany, every second PV systemwith less than 30 kWp is already installed in combination with a similar-sized battery system [32].This results in a complex control problem from an operational viewpoint. Because of thefluctuating nature of the supply of electricity from PV systems, flexibility has to be exploited insuch a way that specific requirements associated with electrical energy, heating energy, and hot-water supply are met while ensuring e ffi cient operation of the overall system. E ffi cient operationcorresponds to profit maximization because electricity can be purchased from the grid or soldto the grid. The control problem must be solved in an integrated manner to accommodate thecomplex dependencies and technical constraints associated with the energy system.The objective of this study is to better understand the optimal operation of the integratedhome energy management system. We consider a single representative residential buildingequipped with a PV system, a modulating heat pump, thermal energy storage systems for floorheating and hot-water supply, and a battery. Based on this building, we analyze the influence ofcommon technological configurations and objective functions on the profit- and energy-relatedkey performance indicators (KPIs). Thus, we identify the structural properties associated withthe optimal operating strategy over a period of one year and investigate the manner in whichthe performance of the overall system is a ff ected by the data forecast horizons. Therefore, weformulate a mixed-integer linear program and analyze the optimal energy flows with a time res-olution of 1 h. We model all the four system components and their interactions and particularlyfocus on the modulating air-source heat pump with distinct modes for floor heating and hot-watersupply. In addition, we provide open-source access to our input data, the modeling parameters,the model, and the visualization code for comprehensibility and reproducibility.The remainder of this paper is structured as follows. Section 2 relates the given controlproblem to the literature. Section 3 introduces the model, and Section 4 describes the modelinputs in detail. Section 5 presents the results of our analysis. Section 6 concludes the studyand provides future research directions. Appendix A presents the complete set of technicalspecifications and the variable and parameter nomenclature, whereas Appendix B presents the2odel formulation.
2. Related Work
Several studies have investigated the design and control of energy systems under variousobjectives. Renaldi et al. (2017) [29] classified the previously conducted studies into two cat-egories: (1) studies conducted using a specific energy simulation software and (2) studies con-ducted using mathematical programming methods. These studies aim to optimize the system’sdesign parameters or operational control under varying objectives. This study will focus on op-timizing and subsequently analyzing the operational control of the underlying energy system.The commonly used domain-specific energy simulation software packages, such as TRN-SYS, ESP-r, Modelica, EnergyPlus, and IDA ICE, provide a detailed physical representation ofthe modeled system and can incorporate nonlinear behavior and stochastic information. How-ever, they are computationally expensive and do not endogenously optimize the system. In ad-dition, the simulations are often di ffi cult to establish. Furthermore, the solutions obtained usingsuch software are not generic but customized to the underlying energy system.In contrast, mathematical programming methods use simplified models to generate a moregeneric—but less physically accurate—solution using reduced-complexity models. The formu-lation of heat pump models has been studied by Bloess et al. (2018) [2]. These models canbe used for optimizing the design parameters of the system or operational control. Further, acomplex problem can be converted into a computationally tractable problem via simplifications.For example, the underlying demand profile is mostly modeled using external tools or simplecorrelations [29]. The accuracy and computational e ff ort must be appropriately balanced in caseof an e ff ective model [27, 15]. The appropriate complexity levels and model formulations of heatpump systems in di ff erent application contexts are presented by Clauss and Georges (2019) [8].Subsequently, we will focus on the mathematical programming approaches. Figure 1: Classification of the control approaches for heat pumps (adapted from Fischer and Madani (2017) [15]). Thelight blue boxes indicate the classification hierarchy of this study.
Herein, we aim to analyze the operational control of a heat pump and PV system. Fig-ure 1 shows the comprehensive taxonomy of this problem suggested by Fischer and Madani(2017) [15]. They distinguished between predictive and nonpredictive approaches. The non-predictive approaches include rule-based and schedule-based algorithms, which are based on3xplicit decision rules. Thus, the heat pump is activated when the room temperature, time, orsome other system state reaches the specified value.In contrast, predictive control approaches determine the operation of a heat pump based onthe forecasted demand, renewable energy generation, and prices. Further, predictions can be gen-erated internally based on past observations, or using external information sources. Uncertaintiescan be introduced into the analysis by varying the accuracy degree of the forecast. However, moststudies assume perfect forecasts with no mismatch between the values known to the optimiza-tion model and those observed by the controlled system [15]. In such cases, optimal controlis evaluated using the optimization model, and real-world applicability may only be partiallyguaranteed. It is still unclear whether this gap can be closed by more realistic (simulation) mod-els or advanced methods of transfer learning, which transfer knowledge from one (potentiallysimplified) model to another model or the real world.The predictive control approaches can be further di ff erentiated into model-based and model-free decision algorithms. The model-free decision algorithms consider the forecasts and imple-ment heuristics based on expert knowledge or learn by interacting with the environment (e.g.,reinforcement learning systems). The model-based decision algorithms, commonly known asmodel predictive control (MPC) algorithms, formulate a physical world model and solve it us-ing an exact or approximate mathematical optimization approach. A high-performing predictiveapproach potentially requires a large implementation e ff ort [15]. In this study, we propose apredictive model-based decision algorithm by assuming that perfect information can be obtainedfrom an external source.Heat pump control can be categorized based on its modeling properties and targeted ap-plication (grid-based, price-based, or renewable-energy-based) [15]. This study belongs to therenewable-energy-based category because it predominantly analyzes the impact of integrating aheat pump with a PV system. Additional objectives include o ff ering services to the grid or ben-efiting from dynamic pricing schemes. Further, we summarize the studies on integrated homeenergy management systems, which are most closely related to our study.Salpakari and Lund (2016) [31] proposed a deterministic dynamic programming algorithmfor the energy management of a low-energy house in Finland. This house was equipped with aground-source heat pump, a PV system, a thermal storage system, batteries, and shiftable loads.They compared the performances of a rule-based self-consumption-maximizing algorithm andcost-optimal control with and without grid feed-in. They analyzed one years’ worth of data usingthe rolling horizon approach with a prediction and control horizon of 24 h and a time resolutionof 1 h. They observed that the shiftable loads o ff er less flexibility than the thermal and batterystorage systems.Vrettos et al. (2013) [38] proposed a deterministic MPC algorithm for the energy manage-ment of a residential building containing a PV system, a battery, and an air-source heat pump.The demand for hot water was fulfilled using an additional electric heater. Their study focusedon the cost reduction potentials based on demand-side flexibility and dynamic prices. By usingthe load data observed from an exemplary week in April, they observed that the energy manage-ment systems operated using the day-ahead or online price information can be beneficial for thegrid and provide the owner with considerable profits. They formulated a quadratic problem andsolved it using the rolling horizon approach with a prediction and control horizon of 16 h and atime resolution of 1 h. They assess the annual electricity costs and grid dependency for di ff erentscenarios and compare the results with those obtained from a rule-based benchmark. The gridfeed-in was not considered in this case.Fischer et al. (2017) [16] formulated a quadratic problem for the energy management in case4f a multi-family building. Their system included a PV system and an air-source heat pump withthermal storage units. They evaluated di ff erent rule-based and MPC algorithms in a dynamicsimulation with an accurate heat pump model but did not provide the prediction and controlhorizons of the model. They modeled a set of virtual heat pumps and applied postprocessingto account for the nonlinear coe ffi cients of performance, the separate operation modes of floorheating and hot-water supply, and the minimum compressor speeds. Further, they identified thetrade-o ff s between thermal losses, the operational e ffi ciency of the heat pump, and the operatingcosts. Their proposed MPC consistently outperformed the rule-based algorithms, although thegap could be significantly reduced by ensuring thorough calibration of the rules. In addition, theirresults were insensitive to the forecasting errors. Today’s values could be reasonably estimatedbased on yesterday’s values because of the high correlations between consecutive days. Theiranalysis did not include a battery and grid feed-in, and the model and simulation runtimes werenot presented.Our approach adopts and extends the heat pump model of Dengiz et al. (2019) [9]. They pro-posed two models, among which one minimizes the heating expenses without considering theelectricity demand and PV generation and the other maximizes self-consumption while consid-ering PV generation. However, both of these models did not incorporate grid feed-in or batterystorage. They designed rule-based heuristics for a home energy management system with dataprivacy. Using scaled-up versions of the aforementioned models, they conducted a numericalstudy with respect to a residential area containing 40 buildings. They employed the rolling hori-zon regime with a prediction and control horizon of 24 h and a time resolution of 5 min. TheKPIs of di ff erent algorithms for the two objectives are compared over 12 separate weeks duringthe heating period. The structure of the optimal energy flows and seasonal patterns were notpresented.After reviewing the previous studies, we identified the requirement for in-depth analysis anddiscussion of the optimal control of an integrated home energy management system. Until now,the impact and calibration of the essential model elements (the objective function and compo-nents, including the grid feed-in and the prediction and control horizons) remain unclear. Inaddition, the model runtimes are required to balance the accuracy and computational e ffi ciency.We hope to foster further research in the field by providing an open-access dataset and modelto the community, resulting in improved control algorithms, new application scenarios, and ad-vanced policy recommendations. These issues will be discussed in the following sections.
3. SHEMS: Model Formulation
In this study, we consider the smart home energy management system (SHEMS) of a singleresidential building (Figure 2) with a time resolution of 1 h. The system includes a PV sys-tem ( pv ), battery ( b ), and dual-mode modulating air-source heat pump ( hp )). At any time, theheat pump can supply heat to the floor heating system ( f h ) or the hot-water system ( hw ). Thefloor-heating temperature and hot-water volume must be maintained within a certain comfortrange. Further, we consider two types of thermal energy bu ff ers: the built-in thermal mass of thefloor heating system and a domestic hot-water tank. The thermal bu ff ers and battery su ff er fromdissipation losses. The building is connected to the power grid ( gr ), enabling the purchase ofadditional electricity and selling of surplus electricity.The known parameters include the outside temperature ( t outside ), PV generation ( g e ), the de-mand for floor heating ( d f h ), hot water ( d hw ), and electricity ( d e ), and tari ff s for purchasing fromthe grid ( p buy ) and selling to the grid ( p sell ). The tari ff s are assumed to be constant. In addition,5he amount of exchange between the grid and building is not restricted in any direction. In thisscenario, directly selling electricity from the PV system to the grid is always better than redi-recting flows via battery, which results in conversion losses. Therefore, we omit the interactionbetween the battery and the grid. Furthermore, simultaneous charging and discharging of the bat-tery is suboptimal because of conversion losses. Other circumstances may apply when arbitragee ff ects are introduced based on dynamic prices or peer-to-peer trading. Figure 2: Energy flows of the smart home energy management system ( the parameters within the dotted frames areexogenous but known to the model).
This study intends to determine the cost-optimal energy flows and heat pump operation dur-ing each hourly time period. We allow violations of the comfort ranges of the floor-heatingtemperature and hot-water volume but impose virtual costs on discomfort to prevent infeasiblesolutions, which can occur in a monovalent system with no additional backup resistive heater.Similarly to Masy et al. (2015) [22], we combine the two objectives using a weighted sumapproach.We formulate the problem as a mixed-integer linear program (MILP). MILPs are popularlyused because of their computational e ffi ciency. We can holistically consider the structural prop-erties of the optimal control policy by setting a prediction and control horizon of one year with atime resolution of 1 h.The system of equations is presented in Appendix B. We describe the model componentsin detail in the following subsections. Section 4 presents the inputs of the numerical study. Theparameter settings and technical specifications are detailed in Appendix A. The net profit of the model can be maximized using the objective function of the model(Equation B.1) by balancing the revenues obtained by selling to the grid with the cost of sourcingfrom the grid. In addition, there is a third term that considers the violations of the comfortablerange of floor-heating temperatures and hot-water volumes. Comfort violations occur when thestates of charge (SOCs) exceed the upper or lower bounds of the comfort range. Following6 weighted sum approach, we add the violations weighted by the parameter cost f actor (seeSubsection 4.7 for details).In our case, all positive and negative violations of the floor-heating constraint can be avoidedby installing air-conditioning and a backup heater, respectively, as implemented in a previousstudy [16]. Negative violations can also be resolved by increasing the capacity of the heat pumpor the thermal storage system. However, designing a balanced system is beyond the scope of thisstudy.
Equation B.3 ensures that the electricity demand ( d e ) is fulfilled at any time, either from thePV system ( X pv → d e ), battery ( X b → d e ), or grid ( X gr → d e ). Meanwhile, Equation B.4 ensures that thesum of flows from the PV system toward the demand ( X pv → d e )), battery ( X pv → b ), grid ( X pv → gr ),and heat pump ( X pv → hp ) is equal to the overall electricity generated ( g e ). The technical specifi-cations of the PV system are presented in Table A.8.We assume that buying from and selling to the grid is always possible. This model could alsobe adapted for configuring and operating an island grid; however, this adaptation is beyond thescope of our study. Equation B.5 can be used to determine the energy balance of the battery SOC (
S OC b ). S OC b increases with the increasing electricity supply from the PV system ( X pv → b ) after considering theconversion losses η b . It deceases when the battery supplies the electricity demand ( X b → d e ) or heatpump ( X b → hp ) (with an e ffi ciency loss of η b ). We also introduce a small dissipation loss ( loss b )proportional to S OC b . This self-discharge is normally neglected because it is only a few percenteach month [37]. However, in our case study, this self-discharge is considered for standardizingthe daily load patterns. If charging at an earlier time in the day is disadvantageous, the chargingtime is shifted to as late as possible on the same day without significantly a ff ecting the KPIs. S OC b is bounded by soc minb and soc maxb , which are obtained based on the specified usablebattery capacity (Equation B.6). Equations B.7 can be used to obtain the maximum charg-ing–discharging rate ( b maxrate ). Simultaneous charging and discharging can be avoided by the coststructure and dissipation losses. The maximum charging rates are determined based on the spec-ified nominal power of the inverter b max . The technical specifications of the used battery arepresented in Table A.10. To model the integrated heat pump system, we must understand its interactions with theoverall home energy management system (Figure 2). The electricity for the heat pump can besupplied by the PV system ( X pv → hp ), the grid ( X gr → hp ), or the battery ( X b → hp ) (see Equation B.8).The dual operation modes are represented by separate energy flows X hp → f h and X hp → hw .Equations B.10 ensure that only one mode can be active at any given time. Therefore, we in-troduce the binary variable HP switch , which is set to one and zero when the heat pump is in thefloor-heating mode and hot-water mode, respectively. Equations B.9 restrict the heat pump loadto its maximum power ( hp max ). Here, the modulation degrees of the floor-heating and hot-watermodes, denoted as Mod f h and
Mod hw , respectively, are continuous variables between zero andone. 7n accordance with Dengiz et al. (2019) [9], we model the thermal energy state of chargewith respect to the mass of floor heating based on its temperature T f h . The floor-heating energybalance is then modeled using Equations B.11– B.13. The next T f h ( h +
1) is equal to the sum-mation of the current T f h ( h ) with the energy supplied by the heat pump X hp → f h multiplied by thecoe ffi cient of performance cop f h , minus the heating demand d f h and the dissipation losses of theheating system Loss + / − f h .The coe ffi cient of performance cop f h ( h ) describes the ratio of the thermal power output to theelectrical power input (see Subsection 4.5 for details). The loss term is that defined by Dengizet al. (2019) [9] with slight modifications, i.e., the sign of the loss f h depends on whether theoutside temperature t outside is greater than or lower than T f h . If the outdoor temperature exceedsthe indoor temperature, the binary variable Hot becomes one (Equations B.14). In this case, the
Loss + / − f h in Equations B.13 becomes negative and the loss f h heats the system instead of coolingit. Although this binary setting simplifies the real-world situation, it prevents heating in summerthrough depreciation losses. All the factors influencing T f h must be multiplied by a conversionfactor conv f h that converts kWh to ◦ C. The conversion factor, given by Equation B.12, requiresthe hourly resolution and volume specification v f h associated with floor heating (see Dengiz etal. (2019) [9] for details).Equations B.15 determine the comfort-range violations associated with the floor heating sys-tem. A positive comfort violation T + f h occurs when the SOC T f h in a given time period exceedsthe upper bound of the comfort range t maxf h . Conversely, a negative violation T − f h occurs when T f h is below the lower bound of the comfort range t minf h .The hot-water constraints (Equations B.16–B.18) are governed by the same principles. How-ever, the loss factor loss hw is a positive constant, and the conversion factor conv hw . convertskWh to liters. The conversion factor, given by Equation B.17, requires the hourly resolution andsupply temperature t supplyhw of the hot-water tank (see Dengiz et al. (2019) [9] for details). Thecomfort violations (Equations B.18) are given not in ◦ C but in liters because the comfort rangeis given as the volume of well-tempered hot water. In the objective function, these units areconverted to EUR based on the cost factor.The model formulated using Equations B.1– B.18 is an extended variant of the multiperiodcapacitated transshipment problem. Furthermore, it includes binary decision variables that en-sure single-mode operation of the heat pump and sign adjustment of
Loss + / − f h . Thus, the problembecomes a MILP that can be e ffi ciently solved under the given problem sizes.
4. SHEMS: Data Input
We selected open-access data sources and open-source modeling frameworks to ensure thereproducibility of our model. In this section, we describe our model data and the adopted tech-nical specifications.
The model is written using the mathematical optimization language
JuMP (package ver-sion 0.21.2) under the
MPL License [11]. The package is embedded in the
Julia programminglanguage (version 1.4.0) under the
MIT License . Both these software packages are open-sourceand available free of charge. The implementation code, data, and visualization code are avail-able on
GitHub under the
MIT License . The model is solved using the
Gurobi solver (packageversion 0.7.6) under an academic license. 8 .2. Heating, Hot Water, and Electricity Demand
To the best of our knowledge, no publicly available dataset contains the electricity demandsand thermal loads associated with heating and hot water usage in single households. Accord-ingly, we constructed a dataset using
BEopt–Building Energy Optimization with Hour-by-HourSimulations provided by the
National Renewable Energy Laboratory (NREL). The software isbased on
EnergyPlus simulations, and were validated by Christensen et al. (2016) [7]. Ourdataset contains the heating-demand, hot-water-demand, and electricity-demands with an aggre-gation level of 1 h. We selected Chicago as the location because the NREL data include onlythe cities in US. Furthermore, the climate conditions in Chicago are similar to those of NorthernEurope (although the solar radiation is slightly higher in Chicago in the cold seasons; comparedto Berlin, autumns are milder and winters are colder). The level of temperatures at this locationis suitable to evaluate the heating control—also in Northern Europe. The aggregated demandvalues, which are comparable to those in Germany [35], are presented in Table 1.
Table 1: Aggregated floor-heating, hot-water, and electricity demands in the given year.
Floor-heating demand Hot-water demand Electricity demand to gen-erate representative results. This bungalow is larger than the average house in Germany (93 m )and smaller than the houses occupied by high-income earners without children (117 m ) andmedium-income earners with children (122 m ). These data are obtained based on the character-istic household types of the German Income and Consumption Sample [3].Because our objective is to control the output of a heat pump in an integrated system, thermaldemands are crucial. Figure 3 shows the hourly mean values of the floor-heating and hot-waterdemands in each month. The floor-heating demand is very high in winter (top row) and almostnegligible in summer (third row). The electricity and hot-water demands remain relatively con-stant throughout the year but follow a distinct daily pattern.
The photovoltaic data were extracted from
Renewable.ninja licensed under
CC BY-NC 4.0 .The PV outputs were derived using the
Global Solar Energy Estimator model [28], which usessatellite observations from the
NASA MERRA reanalysis [30]. The temperature data were ex-tracted from the same sources. Again, Chicago was selected as the reference location to ensurecoherence with the demand profiles.
According to the
Speichermonitoring of RWTH Aachen , in 2017, every second PV sys-tem smaller 30 kWp in Germany has been installed in combination with a battery storage sys-tem [32]. This combined system is popularly purchased for several main reasons, includinghedging against the rising electricity prices, supporting the energy transition, or general interestin technology [32]. The average capacity of the installed battery (7.8 kWh in 2017) is trending9 [ k W h ]
12 1 2 [ k W h ] [ k W h ] time [ k W h ] time time d fh d fanfh d hw d e Figure 3: Electricity, floor-heating, and hot-water demand by month (stacked bars). upward, and the unit costs are decreasing. The acceptable cost has remained constant at approx-imately 10,000 EUR, which is the approximate price of our Tesla Powerwall 2 with a capacity of14 kWh (see technical specifications in Table A.10).The ratio of battery capacity (kWh) to the installed PV capacity (kWp) was approximately1.0 in 2017 and has increased with the increasing size of the systems. Many residential PVsystems in Germany are smaller than 10 kWp (below this limit, the renewable energy levies onself-consumption are avoided and users can profit from the high feed-in tari ff ) [4]. As of January2020, new installations are guaranteed a feed-in tari ff of 9.87 ct / kWh over a time horizon of20 years. In 2017, the mean installation size was 8.1 kWp and was continuously increasing. Weselected a 10 kWp system because it is cost-e ffi cient and accounts for the purchase motivationsand growth trends. The specifications of the selected heat pump are presented in Table 2. This heat pump wasclassified using the framework presented by Fischer and Madani (2017) [15]. Specifically, we10odel a modulating (variable speed) heat pump that can be continuously adjusted (rather thana heat pump that can be merely switched on and o ff ). The designed air-to-water heat pump hastwo separate storage units for floor heating and hot water. Here, the thermal building mass of thefloor heating system provides the storage volume for floor heating.The air-source heat pump was selected because it comprises approximately 82% of all thenewly installed heat pumps in Germany. This type of heating is typically combined with a floorheating system in newly constructed buildings.Our objective is to elucidate the structure of the optimal control strategy of an integrated sys-tem. We excluded the minimum run time, part load performance, and cycling losses to simplifythe model as far as possible. Although these simplifications are in agreement with the scope ofthis study [21], they impose various limitations, as discussed in the concluding remarks. All thetechnical specifications of the heat pump are presented in Table A.9. Table 2: Heat-pump characteristics [15].
Heatsource Heatsink Heatdistribution Heatstorage Heatsupply concept Capacitycontrol
Air Water Underfloor Tanks Monovalent Modulated
Coe ffi cient of Performance of the Heat Pump. The coe ffi cient of performance ( cop ) de-termines the ratio of the thermal power output to the electrical power input. The cop of heatpumps is often simplified and considered to be a constant (typically, cop = cop by considering the temperature di ff erenceand compressor speed. They applied Taylor linearization on virtual heat pumps. In accordancewith Dengiz et al. [9] and using manufacturers’ data, we approximate the cop based on the di ff er-ence between the sink and source temperatures. The evaluated parameter, given by Equation 1,is exogenous to the model, and the model remains linear. cop f h / hw ( h ) = max (cid:40) . − ∗ | t supplyf h / hw − t outside ( h ) | , (cid:41) (1)In accordance with Dengiz at al. (2019) [9], we set the supply temperatures of the floor-heating and hot-water systems to t supplyf h = ◦ C and t supplyhw = ◦ C (using a fresh water station),respectively. As shown in Figure 4, cop increased in summer because the gap between thesupply and outside temperatures decreased. High coe ffi cients of performance can be achieved inhigh-insulated buildings with floor heating, where a low temperature supply is su ffi cient. ff s Under the current regulations in Germany, most customers do not pay dynamic electricitycosts or trade at the day-ahead or intraday market. Thus, our model assumes a constant PV feed-in tari ff of 10 ct / kWh and a constant electricity retail price of 30 ct / kWh. These values are basedon the current feed-in tari ff and current end-consumer retail prices of green electricity [4, 17, 23].In Germany, the guaranteed feed-in tari ff s are continuously decreasing [4]. Although thePV production costs have reduced to become approximately one-third of the retail electricityprices [18], the guaranteed feed-in tari ff barely covers the production costs. Thus, feeding one’sPV production into the grid is not profitable. Profitability can be achieved only by increasingone’s share with respect to the local energy consumption (see Subsection 5.1).11 month . . . . . . . . c o e ff i c i e n t o f p e r f o r m a n c e − − [ ◦ C ] cop fh cop hw t outside Figure 4: Coe ffi cients of performance of the heat pump for floor heating (and hot water in a given year)(Points: hourlymean values, Lines: monthly median values, Areas: 50% confidence interval, Black Line: mean outside temperatures). The cost factor parameter in the objective function weighs the importance of the comfort vi-olations with respect to the net profits. This trade-o ff can be modeled using di ff erent approaches.Pean et al. (2019) [27] comprehensively reviewed various approaches. Increasing the cost factorreduces the profits and the total number of comfort violations (Figure 5). Based on our prelimi-nary experiments and a previous study [40], we observed that a symmetric linear penalty functionwith a cost factor of 1 gives Pareto-optimal results.Therefore, both the positive and negative violations of the comfort ranges with respect tofloor heating and hot-water supply were equally penalized at marginal costs of 1 EUR. Increasingthe cost factor further, as in [22], does not reduce the cumulative number of comfort violationsbecause such comfort violations can be solely attributed to the lack of air conditioning in summer.Based on the model in B.1 to B.18 and the aforementioned parameter specifications, weconstructed a fully specified instance of a representative residential building with the requiredtechnical equipment combination. In the following section, we will analyze the optimal behaviorof the system and discuss improvements of integrated control.
5. Results and Discussion
In this section, we conduct an extensive numerical analysis to understand the optimal be-havior of SHEMS. Further, we investigate the potential of combining all the technological com-ponents into a single integrated control. In addition, we scrutinize the structure of the optimalcontrol policy under seasonal e ff ects and assess the value of information. Then, we discusspromising mechanisms for improving the rule-based control schemes.In Subsection 5.11, we examine the common KPIs associated with four di ff erent technolog-ical configurations and two alternate objective functions. In the first case study, the e ff ects ofintegrating a battery and the functionality of grid feed-in into the system are examined. In the12 − − cost factor p r o f i t s [ E U R ] c o m f o r t v i o l a t i o n s comfort violationsprofits Figure 5: Numerical experiment determining the cost factor in the objective function (base case). second case study, two objective functions are evaluated: maximizing the self-consumption andmaximizing the self-su ffi ciency. In Subsection 5.2, we discuss the optimal energy flows through-out the year to identify seasonal patterns in demand fulfillment under di ff erent conditions. Theflow analysis is supplemented by the evaluation of the intraday e ff ects based on illustrative edgecases during summer and winter. To design new decision rules, one must determine target orthreshold values. Therefore, in Subsection 5.3, we analyze the target SOCs, i.e., the levels tocharge up to, of the storage systems and the schedules, i.e., the time at which charging is opti-mally conducted. Finally, the value of information in the rolling horizon planning approach isdiscussed in Subsection 5.4. We determine the e ff ects of adapting prediction and control timehorizons, i.e., the foresight capability of the model and the number of fixed time periods in eachexecution step, respectively, on the overall performance of the model. The KPIs of our model were evaluated for four technological configurations, hereafter calledcases, and three objective functions (Table3).In the base case (Case 1), the system was established based on the specifications in Section 4.The system did not contain a battery in Cases 2 and 4. In Cases 3 and 4, the feed-in tari ff waszero. In all the four cases, we applied the objective function described in Section 3 (Profit).Further, we determine the e ff ects of maximizing the self-consumption (Objective 2) and self-su ffi ciency (Objective 3). These objective functions are commonly used in literature [15].Luthander et al. (2015) [20] defined self-consumption as the share of locally produced elec-tricity consumed locally. We minimize feeding into the grid to incorporate this goal in ouradapted objective function (Equation 2). min (cid:88) h ∈ H (cid:16) X pv → gr ( h ) + C violation ( h ) (cid:17) (2)Self-su ffi ciency (or autarky) is defined as the share of local demand satisfied by local pro-duction. The reference point in this case is not the renewable generation but the local demand.13 able 3: Configurations and objective functions of the numerical study Technological configurations (objective function = Profit)
Case 1 Base case Battery installed and grid feed-in allowedCase 2 No battery No battery installed but grid feed-in allowedCase 3 No feed-in Battery installed but no grid feed-in allowedCase 4 No both No battery installed and no grid feed-in allowed
Objective functions (technological configuration = Base case)
Objective 1 Profit Minimize the net cost of grid exchange and comfort viola-tionsObjective 2 Self-consumption Minimize the grid feed-in and comfort violations (sum ofviolations is constrained to the base case value)Objective 3 Self-su ffi ciency Minimize sourcing from the grid and comfort violations(sum of violations is constrained to the base case value)Accordingly, Equation 3 minimizes sourcing from the grid. min (cid:88) h ∈ H (cid:16) X gr → d e ( h ) + X gr → hp ( h ) + C violation ( h ) (cid:17) (3)In both the functions, we retain the original penalty terms associated with the comfort viola-tions. However, Equation 4 constrains the sum of comfort violations with respect to the optimalvalue of the base case ( violations bc ), determined as 79 units in our numerical experiments, tocomply with the multiobjective setup and ensure comparable results. (cid:88) h ∈ H (cid:16) T + f h ( h ) + T − f h ( h ) + V + hw ( h ) + V − hw ( h ) (cid:17) ≤ violations bc (4)We must prevent simultaneous charging and discharging of the battery in the alternate ob-jective functions. In the base case, simultaneous charging was circumvented based on the coststructure and the dissipation losses of the battery. We replace Equation B.7 by Equations 5 foranalyzing the alternate objectives. The binary variable B switch ( h ) is switched on and o ff when thebattery is charged and discharged, respectively, preventing simultaneous charging and discharg-ing: X pv → b ( h ) ≤ B switch ( h ) ∗ b maxrate , ∀ h (5) X b → d e ( h ) + X b → hp ( h ) ≤ (cid:16) − B switch ( h ) (cid:17) ∗ b maxrate , ∀ h The annual aggregated results are summarized in Table 4. The overall annual energy con-sumption includes the electricity usage that fulfills the demand de and charges the thermal storagesystems. We also report the self-consumption and self-su ffi ciency rates.The overall profits refer to the net sum of the revenues and costs of grid exchanges. Weexclude the virtual costs associated with comfort violations to improve the interpretability. InCases 3 and 4, where the grid feed-in tari ff was zero, PV generation could not always be pro-cessed by the system; in particular, the storage systems could not be utilized when the supply14 able 4: Annual KPIs of the numerical study (Seco: self-consumption, Sesu: self-su ffi ciency). Case 1: Case 2: Case 3: Case 4: Obj.2 Obj.3Objective: Profit Profit Profit Profit Seco SesuBattery: Yes No Yes No Yes YesGrid feed-in: Yes Yes No No Yes YesEnergy consumption [kWh] 7515 7517 7590 7562 7701 7592Self-consumption rate [%] 37 24 38 24 39 38Self-su ffi ciency rate [%] 79 52 79 52 79 79Overall profits [EUR] 570 183 -474 -1085 517 557PV curtailment [kWh] - - 10309 12641 - -Sum of violations [ ◦ C] or [l] 79 79 80 79 79 79Runtime [min] 47 39 14 17 36 10Mean
S OC b [kWh] 2 - 3 - 7 4Mean T f h [ ◦ C] 21 21 21 21 21 21Mean V hw [l] 99 99 78 88 97 92exceeded the demand. This is referred to as PV curtailment . Further, we report the sum of the vi-olations of the upper and lower bounds with respect to the comfort ranges for hw and f h . Finally,we present the runtimes and average SOCs of the storage systems. Battery and Grid Feed-in.
Massive PV curtailment could be observed in the system inthe absence of grid feed-in (Cases 3 and 4). The system could not locally utilize the PV surplusdespite the optimized control and thermal storage. In Case 3, this e ff ect was partly mitigated bythe battery; therefore, the self-consumption, self-su ffi ciency, and profit were higher than thosein Case 4. The energy consumption was slightly improved ( < ff ect on the self-consumption and self-su ffi ciencyrates.In addition, feed-in did not significantly a ff ect the levels of comfort violations. Inherentviolations of the temperature comfort range were observed during the summer because of no airconditioning.The grid feed-in significantly changed the system behavior by monetizing the PV surplus.Therefore, it should be incorporated into the integrated energy management system if availablein the considered market. Alternate distribution channels, such as peer-to-peer networks, shouldbe installed in markets with no feed-in tari ff or when tari ff s are decreasing. Thus, the PV batterysystems of a given size can be e ff ectively utilized. The battery increases the profits by increasingthe self-consumption and (especially) self-su ffi ciency rates.The overall system performance is considerably improved using the integrated approach.However, the runtime of the model increases when considering the feed-in tari ff s. In the base15ase, the runtime required for optimization throughout the year was approximately 50 min, al-most all of which was consumed by the solver. The runtime may be significantly reduced byincorporating the initial solutions during subsequent runs. In Subsection 5.4, the solution timecan be reduced using the rolling horizon approach. Self-Consumption and Self-Su ffi ciency. The optimization of self-consumption (Objec-tive 2) slightly increased the self-consumption ( + ffi ciency objective (Objective 3) achieved better results than Objective 2. How-ever, the self-su ffi ciency rate did not increase, whereas the energy consumption increased by 1%and the profits decreased by 2% compared with those observed in the base case. Local PV gener-ation is preferred over sourcing from the grid, which incurs depreciation losses. In the base case,the e ffi ciency could be increased by selling to the grid now (to avoid storage losses) and sourcingfrom the grid later. The runtime with self-su ffi ciency optimization was 79% lower than that inthe base case and was even lower than those associated with the reduced-technical setups.Thus, the maximization of self-consumption results in an unintended system behavior. Theself-consumption rates only slightly improved, whereas most of the other KPIs deteriorated.Maximizing the self-su ffi ciency did not significantly change the base case results except for theruntime, which was considerably reduced. Meanwhile, our initial cost-minimizing objectivefunction improved the e ffi ciency which results in reduced energy consumption and increasedprofits. In this subsection, we analyze the optimal flows in the base case (Case 1; with a battery andgrid in-feed) combined with our initial cost-minimizing objective function. This analysis willenhance our understanding of the optimal behavior of the integrated system.
Seasonal Flows.
Figure 6 illustrates the seasonal mean PV generation (yellow line), thecorresponding sinks (stacked bars), and the mean battery SOC (purple line). During summer, thelow heating demand and high PV generation naturally increased the grid feed-in (gray bars inFigure 6). The lower battery SOC in summer when compared with the remaining seasons willbe analyzed in Subsection 5.3. Generally, the battery charging was delayed toward the end of theday because of self-discharging losses. The exceptions to this rule indicate that both seasonalityand time of day (both how much and when to charge) are important in rule-based approaches.However, Figure 6 shows only aggregated seasonal flows and does not allow the analysis of the16 time [ k W h ] winter time spring time summer time autumn g e SOC b X pv → hp X pv → gr X pv → b X pv → d e Figure 6: Mean flows from the PV system (stacked bars) and battery SOCs (purple lines) during each season (base case). daily operational patterns. In the next two paragraphs, we analyze the operational patterns ofsome edge cases observed during winter and summer.
Winter Edge Cases.
The winter season is especially challenging because of the high heat-ing demand and low PV generation. Figure 7 shows the behavior of the system around the winteredge case (January 4–6). The PV production was exceptionally low on January 4.The PV flows and battery SOCs (top row of Figure 7) are similar to those observed in Fig-ure 6. The dotted line denotes the maximum usable battery capacity. The second row of Figure 7plots the electricity demand and the sources fulfilling that demand (stacked bars). The third rowdenotes the thermal demands (areas under the curve), the sources of the heating energy (com-prising stacked bars), and the modulation degree of the heat pump (bar heights). The dotted lineindicates the maximum power of the heat pump. The cop compensates the di ff erence betweenthe electrical and thermal energy. The bottom row illustrates the e ff ect of the heat-pump oper-ation on thermal SOCs. Further, the mode of the heat pump is indicated, i.e., whether it is o ff or running in the f h or hw modes. The left axis indicates the temperature of the f h system (redline), whereas the right axis indicates the available volume in the hw tank (dashed blue line). Thedotted lines indicate the comfort ranges of both the systems.On January 4 (column 1), the PV generation is insu ffi cient to fulfill the electricity and thermaldemands (rows 2 and 3, respectively). Hence, sourcing from the grid is necessary even duringday (gray bars).On January 5, both the thermal and electricity demands were fulfilled by PV generation.Subsequently, the battery was not fully charged (purple line in row 1) because the maximumnominal power of the inverter was reached 10–12 h. The PV surplus was fed into the grid (graybars in row 1).On January 6, the demands were fulfilled and the battery was fully charged. At 9 h, some gridfeed-in was allowed. Again, the grid feed-in at 10–13 h can be attributed to the power restrictionsof the inverter. Charging was delayed as long as possible owing to the self-discharging losses ofthe battery. During the nighttime hours of all the days, the battery power was insu ffi cient andenergy was sourced from the grid to fulfill the demand. Summer Edge Cases.
Figure 8 presents the flows from the PV system and the electricityand thermal demands on three days in summer (August 11–13) when the thermal demand was17 [ k W h ] g e SOC b X pv → hp X pv → gr X pv → b X pv → d e . . . . . [ k W h ] d e X b → d e X gr → d e X pv → d e [ k W h ] d fh d hw X b → hp X gr → hp X pv → hp time [ ◦ C ] [ L i t e r ] time [ L i t e r ] time T fh Mod fh [ l i t e r ] V hw Mod hw Figure 7: Flows from the PV system and demand fulfillment during winter days (base case). considerably low (row 3). On all the three days, the electricity and thermal demands were ful-filled without sourcing from the grid. The battery was charged to its target SOC (approximately25% of the battery capacity; peaks of the purple line in row 1). The decisive factor in case of thetarget SOC was the electricity demand between the end of PV generation on the current day andthe start of PV generation on the following day (purple bars in row 2). The PV surplus was fedinto the grid.As expected from Figure 3, no floor heating was demanded in summer. The hot-water tankwas heated solely based on PV generation. cop hw peaked when the outside temperature wasthe highest; accordingly, the tank was heated at 13 h on August 11 and 14 h on August 13. Onthe latter day, multiple very small charging instances could be observed immediately beforethe main charging instance. These were necessary to maintain the hot-water level (20 liters)within the comfort ranges (row 4 of column 3). On August 12, when the hot-water demand wasexceptionally low, charging can be avoided (row 4 of column 2). The benefits of the high cop hw on a warm day (August 11) must be counterbalanced against the depreciation losses of the watertank. On most days, charging once a day at the highest cop hw was optimal. The target SOC wasobtained based on the forecasted demand until the next day.Based on these observations and by assuming perfect information, both the rules of the targetSOCs and the best time of the day for charging can be obtained. Considering the cost structureand the dissipation losses associated with the problem, the daily PV flows are fulfilled in the18 [ k W h ] g e SOC b X pv → hp X pv → gr X pv → b X pv → d e . . . . . [ k W h ] d e X b → d e X gr → d e X pv → d e [ k W h ] d fh d hw X b → hp X gr → hp X pv → hp time [ ◦ C ] time time T fh Mod fh [ l i t e r ] V hw Mod hw Figure 8: Flows from the PV system and demand fulfillment on three summer days (base case). following order of priority:1. Electricity demand and target SOCs for heating2. Target SOC of the battery3. Feed-in to the gridThe PV flows can be inferred from the target SOCs. The SOCs and charging times can beeasily derived during summer. Obtaining the target SOCs during winter is more complicatedbecause the solution space is constrained by the low PV generation and high heating demand.The following subsections present an in-depth analysis of the target SOCs.
The target SOC is defined as the maximum value to which the storage should be chargedon a given day. In Figure 9, these daily values are illustrated for the perfect-information run(base case). An SOC is considered as a target SOC if charging was conducted in the previoustime period because the change of state can only occur during the next time period. Therefore,because the floor was not heated in summer, the floor heating graph (Figure 9 (b)) exhibited a lowtrend and less observations during the summer period. Apart from the daily target SOCs (points),Figure 9 plots the monthly medians (lines), 50% confidence intervals (areas), and capacities orcomfort bounds (dotted lines). 19he maximum possible target SOC of the battery (Figure 9 (a)) was defined based on themaximum usable capacity of the battery. Theoretically, the thermal target SOCs are restrictedonly by physical bounds and the penalties imposed on comfort-range violations. Charging isusually scheduled at the times during which peak cop could be observed. month [ k W h ] ( a ) Battery month . . . . . . [ ◦ C ] ( b ) Floor heating month [ l i t e r ] ( c ) Hot water medianboundaries − confidencedays Figure 9: Target SOCs during each month (base case).
Battery Target SOCs.
The target SOCs of the battery exhibited a distinct U-shape (Fig-ure 9 (a)). The median values were close to the capacity limit during winter and decreased tobecome approximately a quarter of the capacity limit in summer. As noted before, this behavioris related to the battery characteristics. To avoid dissipation losses, charging should be conductedin accordance with the immediate upcoming demand (until the next PV generation), and the PVsurplus should be fed into the grid. In winter, the target SOCs were raised accordingly when thefloor-heating demand was high.Observations lower than median values can be attributed to our definition of the target SOC.The reported values were obtained using the optimal solution and were subjected to modelingconstraints. For instance, the battery charge capacity was sometimes limited by the PV gen-eration or inverter capacity (column 2 of Figure 7). In these cases, the observed SOCs werelower than the unconstrained target SOCs. Because these instances mainly occurred in winter,the downward volatility was higher in winter compared with those during the remaining seasons.Observations greater than the median values, which were especially common during the sea-sonal transitions in April and November, can be explained by the diverse floor-heating demandsduring these months. At the beginning of April, the battery is charged to a high target SOC tofulfill the high heating demand. By the end of April, the heating demands have reduced, resultingin lower target SOCs.These results have interesting implications with respect to the design of rule-based controlapproaches. We suggest dynamic control of the target SOC values instead of setting a constanttarget SOC for the battery (as in common practice). These values can be defined based on the ob-served daily target SOC or the monthly or seasonal mean SOCs. The unconstrained target SOCscould be reasonably set to their daily maxima, i.e., their full capacity, to cope with the increasedvolatility observed during November and April. Dynamic target SOCs can potentially reduce thedissipation losses and increase the net profits when compared with the systems having constantvalues without jeopardizing the user comfort. Balancing the trade-o ff between the cost-savingpotential and the information requirements of the alternating temporal aggregations should be20ttempted in future.The aforementioned target SOCs are applicable only to charging from PVs and are inappli-cable to charging from the grid. As explained earlier, when the electricity prices are constant,charging the battery from the grid is economically nonviable because of dissipation losses. Floor-heating Target SOCs.
In Figure 9 (b), the floor-heating target SOCs are concen-trated around the upper bound of the comfort range in winter, whereas they are considerablylower in summer when little or no heating is required. During the transition months, especiallyin May and October, volatility is introduced because of the fluctuating nature of the floor-heatingdemands. Similar to the battery and hot-water behaviors in Figure 8, the floor is heated by PVgeneration when it su ffi ces. On high-demand days, the thermal storage is charged to its capacitylimit.From October to mid-November, the storage was charged (not necessarily to its upper limit)almost daily, preferably at the time of peak cop f h . Smaller charging processes immediatelybefore the main charging process may be necessary to satisfy the comfort constraint.From mid-November to the winter season, the floor was typically heated to the upper-comfortbound and the PVs did not fully cover the heating demand. Therefore, additional power mustbe sourced from the grid to maintain the comfort constraint. The floor-heating control, espe-cially in winter, aims to prevent comfort violations while utilizing the PV generation and battery.Accordingly, on most days, the SOC will be at the lower-comfort bound immediately beforePV generation. When the PV generation starts, it can be fully utilized (see columns 2 and 3 ofFigure 7). Based on these observations, we can derive the floor-heating target SOCs. Hot-water Target SOCs.
Throughout the year, the hot-water demand is more stable thanthe floor-heating demands (see Figure 3) because it is less influenced by the seasonal patterns.Therefore, the hot-water charging behavior is more evenly distributed during the year (Fig-ure 9 (c)).As indicated in Figure 8, hot-water charging depends on the immediate upcoming demand.In summer, the demands on multiple days are sometimes aggregated to exploit the high cop hw on a hot day. In such cases, the target SOC is high (see row 4 in column 1 of Figure 8). TheSOC is lower when the target demand is a single day’s usage (row 4 in column 3 of Figure 8).When the demanded floor heating is high, both the heat-pump modes compete for the maximum cop -hour because only one mode can operate at any one time. During the transition seasons, boththe heat-pump modes attempt to utilize the limited PV generation. This e ff ect slightly increasesthe target values. Especially when the cop value reduces in winter, multiple heating periods arenecessary to achieve a certain target SOC.The observed target SOCs of the storage systems were observed within the bounds of capacityor thermal comfort. Within these ranges, the storage systems were charged under the constraintsof PV generation and (in case of the battery) the inverter capacity. The target SOCs generallycovered the demand up to the next PV generation. In winter, PV generation was insu ffi cient,and the target SOCs were constrained to suboptimal values. Consequently, the target SOCs werewidely diverse and di ff ered on both daily and seasonal time scales. Regardless, they can bederived from the general pattern, and their target values should be set accordingly.Given the important role of the demand values until the next PV generation, we analyze arolling horizon approach with di ff erent prediction horizons in the following subsection.21 .4. Prediction and Control Horizon Until now, we have assumed perfect foresight over the whole time horizon (one year = Figure 10: Rolling horizon approach showing the passed SOCs.
Most previous study assumed a prediction horizon of 24 h [27]. This modeling choice isjustified by the data availability or the argument that long horizons do not significantly improvethe results. The horizon is commonly set from midnight to midnight [31, 9]. In the previoussections, we showed that the optimal charging amount is considerably dependent on the upcom-ing demand until the next PV generation. In the following subsection, we analyze the e ff ect ofrestricted (perfect) foresight. We will evaluate the parameters h predict and h control associated witha rolling horizon approach (see Figure 10).The prediction horizon h predict can be used to define the number of upcoming periods knownto the model. The minimal prediction case is an online system in which the future informationis ignored and fixed rules or schedules are applied to some current state, for example, based onthe outside temperature. Although increasing the prediction horizon improves the model results(assuming perfect information), more data and a more advanced implementation are required.To analyze the value-of-information e ff ects, we applied a rolling horizon approach with di ff erent h predict times (24–96 h). By fixing the control-horizon increment at 24 h, we planned the next h predict , fixed the first 24 h, and set the resulting SOCs at (24 +
1) h as the initial states of the sub-sequent optimization run. This process for h predict =
24 h is illustrated in the top part of Figure 10.To achieve comparable results, the reported evaluation horizon was standardized from January 1to December 27 (8664 h) because h predict = h control defines the number of time periods over which planned decisionsare executed before beginning a new optimization run. The prediction and control horizons areoften uniformly set to 24 h (as in Figure 10 (top)). However, this setting may result in unintended22nd-of-horizon e ff ects. When the whole prediction horizon is fixed at 24 h, the positive SOCsat the end of the horizon add no value to the model. For example, the battery will always beempty at midnight; therefore, it cannot cover the morning demand before the first PV generation.In the bottom part of Figure 10, h control is reduced from 24 h to 6 h while maintaining h predict at24 h. In this setting, the end-of-horizon e ff ect is mitigated when compared with that observed incase of an equal-length prediction and control horizon. To analyze the end-of-horizon e ff ects fordi ff erent values of h predict , we varied h control as 24, 12, and 6 h in the rolling horizon approach. Table 5: Annual KPIs in the rolling horizon approach with varying prediction horizon (base case). h predict h control / year] 7357 7440 7374 7371 7368Self-su ffi ciency rate [%] 80 64 79 79 79Overall profit [EUR / year] 595 360 583 586 592Sum of violations 79 2022 86 82 82Runtime [min] 40 0.13 0.14 0.19 0.48 Prediction horizon.
The results are compared in Table 5. Compared to the full-horizonmodel (8664 h), the rolling horizon approach slightly increased the energy consumption (by 1%at 24 h and 0.1% at 96 h), reduced the self-su ffi ciency rate (by 20% at 24 h and 1% at 96 h),lowered the profits (by 40% at 24 h and 0.5% at 96 h), and increased the proportion of com-fort violations (by 2460% at 24 h and 4% at 96 h). All the KPIs were improved by extendingthe prediction horizon; however, the improvement rates decreased and the runtimes increased.Nevertheless, the runtime (even for h predict = ff ects may also be the cause, asshown in the next subsection.As long prediction horizons require more data, the trade-o ff between the forecasting costsand data quality should be considered along with the overall improvements, and the parametersshould be appropriately set. Table 6: Annual KPIs in the rolling horizon approach with varying control horizon (base case). h predict h control ffi ciency 80 75 78 79 79 79 79 80 80Profit 595 528 571 585 585 588 590 591 591Violations 79 86 86 82 82 82 82 82 82Runtime 40 0.24 0.44 0.29 0.58 0.38 0.77 0.98 2.0123 ontrol horizon. The second important parameter of the rolling horizon scheme is h control .In the previous analyses, the first 24 h was fixed for all the lengths of the prediction horizon.In this scheme, setting h predict =
24 h was disadvantageous, and longer horizons yielded superiorresults. In case of the 24-h prediction horizon, the final hour lacked any foresight (see Fig-ure 10 (top)). Owing to the end-of-horizon e ff ects, the storage systems were always empty at theend of each optimization run. Conversely, the 36-h look-ahead captured not only the final hourof the horizon (24 h) but also the subsequent 12 h foresight, mitigating the negative e ff ects.The end-of-horizon e ff ects (even those of the 24-h prediction horizon) were mitigated byadapting h control . The KPIs of the 24-h prediction case were significantly improved when thelength of the control horizon was reduced. In the 6-h control horizon, the results almost reachedthose of the 36-h prediction case because the final hour (6 h) considers the subsequent 16 h fore-sight (see Figure 10 (bottom)). However, with the increasing lengths of the prediction horizons,the improvement rates decreased, and in the 96-h prediction case no significant improvementswere achieved. Increasing the length of the prediction horizon also extended the runtimes. Theseverity of the end-of-horizon e ff ects and the widely used unfavorable parameter settings neces-sitate further investigations.
6. Conclusions
In this study, we identified the structural properties of the optimal operating policy of theMPC algorithm for a SHEMS in a single residential building. The energy system contains amodulating air-to-water heat pump (maximum power: 3 kW), a PV system (capacity: 10 kWp), abattery (nominal capacity: 14 kWh), and thermal storage systems for floor heating and hot-watersupply. We allow grid feed-in and sourcing by fixing the feed-in tari ff s and retail prices at theircurrent values in Germany.Based on our numerical analysis of four technological configurations, we observed thatthe battery was essential to improve the self-consumption and self-su ffi ciency rates of the sys-tem. Without a battery, solely exploiting the flexibility of the thermal storage system, the self-consumption and self-su ffi ciency rates were only 24% and 52%, respectively. After the installa-tion of a battery, these rates increased to 37% and 79%, respectively. In addition, we observedthat without grid feed-in PV curtailment was unavoidable under the representative demand pat-tern and PV battery system. The grid feed-in improved the processing e ffi ciency of the PV sur-plus, decreasing the overall energy consumption without deteriorating the self-su ffi ciency levelsand causing comfort violations. When feed-in tari ff s are absent or decreasing or when the grid iscongested, additional revenue streams by peer-to-peer trading could be considered. The control,regulation, and fairness of this alternative should be further investigated.A commonly used objective (maximizing the self-consumption) was observed to be eco-nomically nonviable considering the feed-in tari ff s in Germany. The maximization of self-consumption reduced the e ffi ciency of processing PV generation. The slight increase in self-consumption after maximization (from 37% to 39%) was o ff set by increased losses and en-ergy consumption. Moreover, because the feed-in was reduced, the net profits were 9% lowerthan those of cost minimization. The maximization of self-su ffi ciency improved the perfor-mance compared to the self-consumption case and reduced the solution time from approximately50 to 10 min; however, it marginally reduced the profits (by approximately 2%) below those ofcost minimization. The energy consumption decreased compared to the self-consumption casebecause storage losses were considered. Only applying the cost-minimizing objective balancedthe trade-o ff between the e ffi ciency loss and the potential revenue from grid feed-in appropriately.24y analyzing the optimal flows and cost structures, we determined the dominant order inwhich the available PV generation on any given day is distributed within the integrated system.First, the electricity demand was fulfilled, and the heating SOCs were satisfied. The target SOCsdefine the levels to which the storage units should be charged. Next, the battery target SOC wasfulfilled; finally, the PV surplus was fed into the grid. We quantified the target SOCs and deter-mined the best time for charging the storage devices. The target SOCs were determined based onthe immediately upcoming demand until the next PV generation (on the next day). Because thefloor-heating demand was seasonally variable, the battery and floor-heating target SOCs widelyvaried among the summer, winter, and transitional periods; in contrast, the hot-water target SOCsremained constant throughout the year. The times of charging during PV hours were especiallydistinct for heating, charging is mainly determined based on the maximum coe ffi cient of per-formance. Thus, charging is mainly conducted at times of peak outdoor temperatures. Underthe derived rules, the target SOCs can be quantified on daily, monthly, or seasonal time scales,improving the commonly fixed rule-based approaches. The target SOC performances on di ff er-ent time scales should be evaluated via a simulation study based on data obtained from di ff erentyears.By applying a rolling horizon approach, we reduced the solution time from approximately50 min to less than 1 min (obtaining the solution for one year with a time resolution of 1 h).We evaluated the value of information for di ff erent prediction horizons and the end-of-horizone ff ects for di ff erent control horizons. The commonly used prediction horizon (24 h with a controlhorizon of also 24 h) resulted in ine ffi cient system behavior. We showed that the end-of-horizone ff ects are easily mitigated with no further information requirements by reducing the controlhorizon.The following limitations and potential expansions are underway: adjusting the algorithms tohandle the uncertainties in demand and generation [10], investigating more complex heat pumprepresentations [16], developing self-learning algorithms that incorporate stochastic behaviorand nonlinearities and that adapt to changing user behaviors [40, 39], and creating multiagentsystems in which energy communities can trade in a peer-to-peer market [19, 24, 25].By applying the proposed formulations and insights, other researchers can reduce the actionspace associated with complex stochastic algorithms such as those of the self-learning systems.In these model formulations, reducing the decision space can mitigate the curse of dimension-ality in both continuous-state and continuous-action spaces as with modulating heat pumps andcontinuous storage-system charging. 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Flow variables
Demand fulfillment [kWh] X pv → d e ( h ) , X b → d e ( h ) , X gr → d e ( h ) ≥ X pv → hp ( h ) , X gr → hp ( h ) , X b → hp ( h ) ≥ X hp → f h ( h ) , X hp → hw ( h ) ≥ X pv → b ( h ) , X pv → gr ( h ) ≥ States of charge
State of charge of the battery [kWh]
S OC b ( h ) ≥ f h system [ ◦ C] T f h ( h ) ≥ hw [l] V hw ( h ) ≥ Heat pump variables
Modulation degree of f h [%]
Mod f h ( h ) ≥ hw [%] Mod hw ( h ) ≥ HP switch ( h ) ∈ [0 , f h (pos. / neg.) [ ◦ C] T + f h ( h ) , T − f h ( h ) ≥ +/ -) Loss + / − f h ( h ) ≥ hw (pos. / neg.) [l] V + hw ( h ) , V − hw ( h ) ≥ Hot ( h ) ∈ [0 , Additional variables for alternative objectives
Heat pump mode (binary) B switch ( h ) ∈ [0 , able A.8: Specifications of the photovoltaic system (based on [32]) Definition Parameter Value
Capacity of the photovoltaic system pv max
10 kWpSystem loss 0Year 2015Location ChicagoTilt 35 ◦ Azimuth 180 ◦ Table A.9: Specifications of the heat pump system (based on [9])
Definition Parameter Value
Maximal power of the heat pump hp max Floor heating
Capacity of the floor heating system v f h m Density of concrete p concr kgm Heat capacity of concrete c concr kJkg ∗ ◦ C Supply temperature t supplyf h ◦ CLower bound of the comfort range t minf h ◦ CUpper bound of the comfort range t maxf h ◦ CTemperature loss per time period ( +/ -) loss f h ffi cient of performance cop f h ( h ) [8760 × ff erence inside / outside big ◦ C Hot water
Supply temperature of the hot water t supplyhw ◦ CDensity of water p water kgm Heat capacity of water c water kJkg ∗ ◦ C Lower bound of the comfort range v minhw
20 lUpper bound of the comfort range v maxhw
180 lVolume loss per time period loss hw ffi cient of performance cop hw ( h ) [8760 × able A.10: Specifications of the battery system (Tesla Powerwall 2 [36]) Definition Parameter Value
Minimum usable capacity soc minb =
14 kWh) soc maxb ffi ciency (discharging / charging) η b / charging) b max / charging per hour b maxrate loss b Table A.11: Specifications of exogenous data
Definition Parameter Size / Source
Electricity demand [kWh] d e ( h ) [8760 ×
1] [7]PV generation [kWh] g e ( h ) [8760 ×
1] [28]Floor heating demand [kWh] d f h ( h ) [8760 ×
1] [7]Hot water demand [kWh] d hw ( h ) [8760 ×
1] [7]Outside temperature [ ◦ C ] t outside ( h ) [8760 ×
1] [30]
Table A.12: Specifications of tari ff s and cost factor Definition Parameter Value / Source
Retail price (from grid) p buy / kWh [17, 23]Grid feed-in tari ff (to grid) p sell / kWh [4]Cost per unit of comfort violation ( ◦ C / l) cost f actor / Unit [40, 22]30 ppendix B. Mathematical model.Objective function: max (cid:88) h ∈ H (cid:16) p sell ∗ X pv → gr ( h ) − p buy ∗ (cid:16) X gr → d e ( h ) + X gr → hp ( h ) (cid:17) − C violation ( h ) (cid:17) (B.1) where C violation ( h ) = cost f actor ∗ (cid:16) T + f h ( h ) + T − f h ( h ) + V + hw ( h ) + V − hw ( h ) (cid:17) (B.2) s.t.Flow constraints: X pv → d e ( h ) + X b → d e ( h ) + X gr → d e ( h ) = d e ( h ) , ∀ h (B.3) X pv → d e ( h ) + X pv → b ( h ) + X pv → gr ( h ) + X pv → hp ( h ) = g e ( h ) , ∀ h (B.4) Battery constraints:
S OC b ( h + = (1 − loss b ) ∗ S OC b ( h ) + η b ∗ X pv → b ( h ) − (cid:16) X b → d e ( h ) + X b → hp ( h ) (cid:17) /η b , ∀ h (B.5) soc minb ≤ S OC b ( h ) ≤ soc maxb , ∀ h (B.6) X b → d e ( h ) , X pv → b ( h ) , X b → hp ( h ) ≤ b maxrate , ∀ h (B.7) Heat pump constraints: X hp → f h ( h ) + X hp → hw ( h ) = X pv → hp ( h ) + X gr → hp ( h ) + X b → hp ( h ) , ∀ h (B.8) X hp → f h ( h ) = Mod f h ( h ) ∗ hp max , ∀ h (B.9) X hp → hw ( h ) = Mod hw ( h ) ∗ hp max , ∀ hMod f h ( h ) ≤ HP switch ( h ) , ∀ h (B.10) Mod hw ( h ) ≤ − HP switch ( h ) , ∀ h Floor-heating constraints: T f h ( h + = T f h ( h ) + conv f h ∗ (cid:16) cop f h ( h ) ∗ X hp → f h ( h ) − d f h ( h ) − Loss + / − f h (cid:17) , ∀ h (B.11) where conv f h = ∗ p concr ∗ v f h ∗ c concr (B.12) Loss + / − f h = (1 − Hot ( h )) ∗ loss f h − Hot ( h ) ∗ loss f h , ∀ h (B.13)31 f h ( h ) − (1 − Hot ( h )) ∗ big ≤ t outside ( h ) , ∀ h (B.14) t outside ( h ) − Hot ( h ) ∗ big ≤ T f h ( h ) , ∀ hT f h ( h ) ≤ t maxf h + T + f h ( h ) , ∀ h (B.15) t minf h − T − f h ( h ) ≤ T f h ( h ) , ∀ h Hot-water constraints: V hw ( h + = V hw ( h ) + conv hw ∗ (cid:16) cop hw ( h ) ∗ X hp → hw ( h ) − d hw ( h ) − loss hw (cid:17) , ∀ h (B.16) where conv hw = ∗ (cid:16) p water ∗ t supplyhw ∗ c water (cid:17) / V hw ( h ) ≤ v maxhw + V + hw ( h ) , ∀ h (B.18) v minhw − V − hw ( h ) ≤ V hw ( hh
S OC b ( h + = (1 − loss b ) ∗ S OC b ( h ) + η b ∗ X pv → b ( h ) − (cid:16) X b → d e ( h ) + X b → hp ( h ) (cid:17) /η b , ∀ h (B.5) soc minb ≤ S OC b ( h ) ≤ soc maxb , ∀ h (B.6) X b → d e ( h ) , X pv → b ( h ) , X b → hp ( h ) ≤ b maxrate , ∀ h (B.7) Heat pump constraints: X hp → f h ( h ) + X hp → hw ( h ) = X pv → hp ( h ) + X gr → hp ( h ) + X b → hp ( h ) , ∀ h (B.8) X hp → f h ( h ) = Mod f h ( h ) ∗ hp max , ∀ h (B.9) X hp → hw ( h ) = Mod hw ( h ) ∗ hp max , ∀ hMod f h ( h ) ≤ HP switch ( h ) , ∀ h (B.10) Mod hw ( h ) ≤ − HP switch ( h ) , ∀ h Floor-heating constraints: T f h ( h + = T f h ( h ) + conv f h ∗ (cid:16) cop f h ( h ) ∗ X hp → f h ( h ) − d f h ( h ) − Loss + / − f h (cid:17) , ∀ h (B.11) where conv f h = ∗ p concr ∗ v f h ∗ c concr (B.12) Loss + / − f h = (1 − Hot ( h )) ∗ loss f h − Hot ( h ) ∗ loss f h , ∀ h (B.13)31 f h ( h ) − (1 − Hot ( h )) ∗ big ≤ t outside ( h ) , ∀ h (B.14) t outside ( h ) − Hot ( h ) ∗ big ≤ T f h ( h ) , ∀ hT f h ( h ) ≤ t maxf h + T + f h ( h ) , ∀ h (B.15) t minf h − T − f h ( h ) ≤ T f h ( h ) , ∀ h Hot-water constraints: V hw ( h + = V hw ( h ) + conv hw ∗ (cid:16) cop hw ( h ) ∗ X hp → hw ( h ) − d hw ( h ) − loss hw (cid:17) , ∀ h (B.16) where conv hw = ∗ (cid:16) p water ∗ t supplyhw ∗ c water (cid:17) / V hw ( h ) ≤ v maxhw + V + hw ( h ) , ∀ h (B.18) v minhw − V − hw ( h ) ≤ V hw ( hh ) , ∀ hh