MAXIMIZING LOCAL PV UTILIZATION USING SMALL-SCALE BATTERIES AND FLEXIBLE THERMAL LOADS

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1 MAXIMIZING LOCAL PV UTILIZATION USING SMALL-SCALE BATTERIES AND FLEXIBLE THERMAL LOADS Evangelos Vrettos *,1, Andreas Witzig #, Roland Kurmann #, Stephan Koch *, and Göran Andersson * * Power Systems Laboratory, ETH Zurich, Physistrasse 3, Zurich, Switzerland # Vela Solaris AG, Stadthausstrasse 125, Winterthur, Switzerland 1 Corresponding author, phone: , fax: , ABSTRACT: High PV utilization ratios in buildings, i.e. the consumption of most of the PV energy within the building premises, can reduce the energy losses in distribution networs, and mitigate overvoltages and transformer overloadings. For this reason, some countries have adopted, or are discussing, regulatory instruments to motivate local utilization of PV energy. In this paper, we investigate the potential of maximizing the PV utilization ratio using small-scale batteries and flexible thermal loads. We propose four rule-based control algorithms for batteries and heat pumps, and calculate the building energy flows and PV self-consumption ratios on an annual basis using the commercial software Polysun. Based on the results, we provide insights on the potential of thermal and battery storage for PV self-consumption maximization and indicate synergies among them. Also, we identify the battery capacities that maximize the savings over the investment lifetime for different combinations of battery capital costs and PV feed-in tariffs (FITs). The results show that maximizing PV self-consumption can be an interesting business case due to the decreasing trends in battery costs and FITs. Keywords: PV self-consumption, battery, load management, heat pump, control strategy 1 INTRODUCTION 1.1 Motivation Large shares of photovoltaic generation (PV) have been introduced in the power system over the last years, and projections show that the worldwide installed PV capacity will continue to increase [1]. A large portion of the installed PV power is concentrated on the roofs of residential and commercial buildings. Depending on financial incentives, locally produced PV energy can be either self-consumed in the building premises or fed into the grid. Maximizing self-consumption of PV energy is preferable from a technical point of view for the following reasons [2]: (a) less overvoltages occur, (b) cable and transformer thermal limits are less liely to be violated, and (c) the losses in the distribution networ are minimized. The technical advantages of PV self-consumption have been understood by power system regulators in some countries, which have adopted, or are discussing, instruments to motivate local utilization of PV energy in buildings. For example, in Germany the feed-in tariff (FIT) for PV plants has already fallen below the residential customer electricity tariff [3]. This creates an interest for self-consumption of PV electricity even in the absence of regulatory measures, such as the previously existing bonus on self-consumed PV electricity in Germany [4]. However, whether this interest translates to a business case or not needs further investigation. The financial advantages of PV self-consumption need to be taen into account in the planning phase of rooftop PV installations. For this purpose, algorithms to maximize local PV utilization should be incorporated into software for energy simulations in buildings. In this context, this paper presents four control algorithms to maximize PV self-consumption with controllable thermal loads and batteries. To illustrate their applicability, the algorithms were integrated into Polysun, a commercial software for energy simulations in buildings [5, 6], which was extended accordingly by adding new features. 1.2 Related wor Optimal operation strategies for residential and commercial buildings with PV installations and battery storage have recently attracted the interest of many researchers. Some papers, for example [7-10], investigate planning and operation strategies for residential buildings with flexible thermal loads and/or battery storage to reduce electricity costs, but without explicitly addressing the problem of PV self-consumption. Other papers, such as [11, 12], study the potential of PV self-consumption increase using stationary lithium-ion batteries and/or electric vehicles. More relevant to our paper is the wor presented in [13], where active demand side management and storage are compared in terms of potential for PV self-consumption optimization in residential buildings. Event-based loads such as washing machines and dishwashers were considered in [13]; however, heating and cooling loads are neglected. 1.3 Contribution This paper addresses the problem of maximizing PV self-consumption in buildings and its contribution is threefold: (a) four simple rule-based control algorithms for batteries and flexible thermal loads, e.g. heating and cooling loads, are developed and integrated into Polysun; (c) the potential for PV self-consumption maximization is evaluated and the respective cost savings are estimated via annual energy simulations, (d) the effect of uncertain parameters, such as battery costs and FITs, on the optimal building configuration are investigated. Thermal loads in buildings are considered in this wor due to their inherent flexibility. Heating and cooling appliances, such as heat pumps (HPs), air conditioners and radiators, are operated using a hysteresis controller based on a temperature set-point and a dead-band. For heating loads, whenever the temperature falls below the lower dead-band limit, the appliance turns on and eeps heating the room till the temperature reaches the higher dead-band limit. At this point, the appliance turns off and a new cycle begins. To preserve the occupants comfort, the room temperature must be ept within the dead-band. However, the actual on/off state of the appliance at a particular instance is not important. Therefore, the appliance consumption can be shifted in time without noticeable effects on the occupants. In this wor, an HP

2 is considered and the goal is to shift its consumption towards intervals with large PV production. The rest of the paper is organized as follows. Section 2 starts with a brief overview of Polysun and then introduces the models used to simulate the PV, HP, hydronic system, and batteries. Section 3 presents the developed control algorithms for PV self-consumption maximization. The case study considered in this paper is explained in Section 4. Simulation results for building operation during typical wees in winter, spring, summer and autumn are presented in Section 5. Section 6 investigates the effect of thermal storage size on PV selfconsumption and identifies building configurations that maximize the return on battery investment under different assumptions for battery costs and FITs. Finally, Section 7 concludes the wor. 2 MODELING Polysun is a software tool for thermal simulations in buildings developed by the Swiss company Vela Solaris AG. It includes a large number of thermal components and building templates, and allows annual energy simulations using realistic weather and heat demand data. Despite its focus on thermal simulations, Polysun also supports simulations of rooftop PV systems. In this section, we present the mathematical models that are used in this software to simulate the PV, HP and hydronic system. In addition, we summarize recent developments related to the integration of a lead-acid battery model and its internal power management system. 2.1 PV model PV modules are modeled using the general purpose model proposed in [14]. It assumes that the PV is equipped with a maximum power point tracer, and expresses the module efficiency η as a function of solar radiation G and cell temperature T c : o ( G, Tc 25 C) b1 b2 G b3 ln( G), (1) o ( G, Tc ) ( G,25 C) 1 b4 ( Tc 25), (2) where b 1, b 2, b 3 are fitting parameters, and b 4 is the module temperature coefficient. The cell temperature can be calculated based on the ambient temperature T a as follows: G (3) Tc Ta, 1000 where γ is a parameter related to the rear ventilation of the PV module. The minimum set of data needed to identify the four model parameters using linear fitting techniques is: three PV module efficiency values at different irradiance conditions, the PV module efficiency at standard test conditions of 1000 W/m 2 irradiance and 25 o C cell temperature, and the efficiency at 1000 W/m 2 but at a different temperature. Due to its simplicity, the model can be applied for a variety of PV cell technologies including crystalline silicon (csi), amorphous silicon (a-si) and copperindium-diselenite (CIS) cells. The inverter is modeled based on measurement values at 100%, 50% and 10% partial load. Efficiencies for other loadings are calculated applying linear interpolation between these values. 2.2 Heat pump Air-to-water HPs for heating and cooling are considered as flexible thermal loads. The HP model is based on the input values thermal power and electrical power consumption, depending on the temperatures T evap and T air at the evaporator side and air side, respectively. Such values are typically measured according to test standards EN 255 or EN on predefined sampling points (e.g. A2/W35, A2 = air intae temperature of 2 o C, W35 = heating water outlet temperature of 35 o C). Additional sampling points can be used as input values in order to improve the accuracy and application range of the simulation model. A similar approach could be applied for water-water or brine-water HPs. In the time domain simulation algorithm, the operating point of the HP is evaluated at every calculation step. The electrical and thermal HP power, P el and P th respectively, are evaluated by interpolation between the sampling points. These two quantities are also related by the Coefficient of Performance (COP), which is defined as: Pth COP. (4) Pel The resulting HP efficiency can be calculated by: Tevap Tair (5) nc COP. T air 2.3 Hydronic system for heat distribution Thy dynamic system response on the thermal side is highly dependent on the topology of pipes and pumps, on storage tan size, and on controller settings. It has been shown that an adequate representation of the hydronic system is necessary in order to predict the seasonal performance factor of HPs [15]. Reference [16] showed that Polysun algorithms are suitable for this purpose. The included numerical model implements the plugflow approach [17] in which fluid elements such as pipes, heat exchangers, and storage tans realize the mass transport at every single time step. In order to calculate the thermal energy demand, a single-node building model is used and an empirical heat transfer coefficient is applied for the heating appliance, e.g., radiator or floor heating. Note that modeling the building weight and envelope adequately is critical, since they are ey thermal storage elements in addition to the water storage tan. To capture thermal stratification effects, the hot water storage tan is divided into eleven distinct water layers with different temperatures. Numerical weather data from the Meteonorm database [18] are used as external parameters to calculate the building thermal energy demand. 2.4 Lead-acid batteries In this paper, lead-acid batteries are considered due to their low investment cost, which maes them preferable for stationary applications. We are interested in the energy flows among the battery, the loads, the PV and the grid. Therefore, a model that describes the evolution of the stored energy is sufficient for our purposes. On the contrary, purely electrical quantities, e.g., voltages and currents, are not of interest here. The Kinetic Battery Model (KiBaM) is used to simulate the lead-acid batteries [19]. This model considers the battery as a two-well system; the first well contains directly available energy, while the second one contains chemically bound energy, which can be

3 transformed to electricity only at a limited rate. KiBaM accounts for the capacity reduction at increased charge or discharge currents, as well as the recovery effect. In particular, the variant of KiBaM proposed in [20] is implemented in Polysun. This variant assumes that the battery terminal voltage is constant. The available and the bound energy at the end of a charge/discharge interval are given by: t (6) t t ( E0, t c P) (1 e ) P c ( t 1 e ) 1, t 1 1, t, E E e t t t P (1 c) ( t 1 e ) E2, t 1 E2, t e E0, t (1 c) (1 e ), where Δt is the time step duration in hours, E 1,t, E 2,t, and E 0,t are the available, bound, and total energy stored in the battery (E 0,t = E 1,t + E 2,t ), respectively, P is the charge/discharge power, c = E 1,t /E 0,t is the capacity ratio parameter, and is the rate constant parameter that corresponds to the rate at which chemically bound energy becomes available for output. According to our convention, P is positive during discharging and negative during charging. KiBaM also models the maximum discharge (P dis,max ) and charge (P ch,max ) power as a function of the stored energy in the battery according to: t t E1, t e E0, t c (1 e ) (8) Pdis,max, t t 1 e c ( t 1 e ) t t c Emax E1, t e E0, t c (1 e ) (9) Pch,max, t t 1 e c ( t 1 e ) where E max is the nominal battery capacity. With this notation, the State of Charge (SOC) is defined as: E (10) 0, t SOCt. Emax In reality, losses depend on the battery terminal voltage, which in turn depends on the SOC. However, in this paper we are not interested in voltages; for this reason, we represent losses using an aggregate efficiency coefficient. Assuming that charge and discharge efficiencies, n bat,ch and n bat,dis, are equal, the following relation holds for the round-trip efficiency of a complete charge/discharge cycle of the battery (n bat,rt = E discharge /E charge ): n n n (11) bat,ch bat,dis bat,rt. The battery is connected to the AC side via a DC/AC converter, which is simply modeled by its rated capacity and efficiency. The efficiency is assumed to be constant throughout the converter operating range. Battery lifetime estimation is essential to quantify the return of a battery investment. For this purpose, we apply the rainflow cycle counting method [20, 21], which assumes the lifetime to depend on the range and number of charge/discharge cycles. The relationship between the number of cycles to failure (C FL ) and the range (R) of a charge/discharge cycle is modeled via a double exponential function: 3 R 5 R CFL 1 2 e 4 e. (12) Parameters α 1 to α 5 can be directly provided by the battery manufacturer, or obtained via non-linear regression on empirical lifetime test data. We consider twenty bins of the same width and allocate the cycles to them depending on their range. Denote by M i the annual number of cycles with a range R i, and let C FLi denote the respective cycles to failure. In each of these cycles, 1/C FL,i of the entire battery lifetime is consumed. Therefore, the cumulative annual damage (7) can be calculated by: M (13) i D. i CFL, i For instance, if D=0.5 at the end of an annual simulation, half of the battery lifetime has been consumed. In other words, the battery will need replacement every two years. Since battery lifetime is drastically affected by deep discharge cycles, we use only part of the available capacity for daily cycling by introducing a constraint SOC SOC min. The SOC min value can be determined based on manufacturer data. 3 CONTROL ALGORITHMS The PV self-consumption ratio is defined as the ratio between the PV energy consumed in the building premises (E PV bld ), i.e. before the point of common coupling (PCC), and the total PV energy yield (E PV ). In this paper, we assume the meter configuration of Figure 1. Note that instead of M1 and M2, a single bi-directional meter could also be used. Denote by E exp the energy exported to the grid and recorded by meter M1, and by E imp the energy imported from the grid and recorded by meter M2. With this notation, the PV self-consumption ratio can be defined as: E E PV bld PV E exp (14). E E PV PV Figure 1: Layout of the system and metering topology. The controllable components and electricity meters are shown in green and orange, respectively. UL stands for uncontrollable load, i.e. all other load apart from the HP Four simple rule-based control algorithms for HPs and batteries are developed in this paper to maximize ξ. Although more advanced predictive optimization algorithms for building control have been proposed, e.g., in [7], they usually rely on simplified single zone building thermal models, which neglect the thermal distribution system. Instead, in our wor heating appliances and the hydronic system are modeled in detail, which maes model-based control schemes computationally intensive. Our goal is to investigate if even simple control algorithms can significantly increase the PV self-consumption ratio. The ey idea of the algorithms is that PV energy should be utilized as much as possible within the building premises, instead of being fed to the grid. For this reason, HP operation can be shifted to intervals when the PV production is at its maximum. The PV energy is stored as heat in the hot water storage tan and can be utilized later on according to the building thermal energy demand. Similarly, PV energy surpluses can be stored in the battery provided that it is not fully charged. During hours with low

4 or zero PV production, the battery discharges to cover the load reducing the electricity imports from the grid. In this setup, two storage options are available in the building: thermal storage and battery storage. Algorithm 1 (A1) assumes that only the thermal storage is present. Algorithm 2 (A2) considers the battery but does not use the thermal storage for PV self-consumption. Algorithms 3 and 4 consider both storage options but differ with respect to the priority with which they are charged when PV energy surpluses exist. Algorithm 3 (A3) prioritizes the HP, while Algorithm 4 (A4) stores energy first in the battery. The choice of the priority depends on efficiency values, the thermal energy demand of the building, the electricity consumption profile, and it might exhibit seasonal patterns. Since it is generally difficult to tell a priori which priority is the most appropriate, annual simulations can be used for this purpose. The proposed algorithms are compared against a base case (A0) where there is no battery and the HP is operated based on its internal controller. The operation principle of this controller is shown in Figure 2. The water temperatures at the seventh and tenth layer of the storage tan, T s7 and T s10, as well as the building temperature T b, are used as controller inputs. The respective set-points are denoted by T s7,min, T s10,max, T b,min, and T b,max. The HP internal temperature T HP is also used as a control input for reliability reasons, i.e. to allow switching off whenever a limit T HP,max is exceeded. In the following, the HP state is denoted by S HP and (t) denotes the current time step. According to Figure 2, the HP state changes when any of the above set-points is reached. In any other case, the HP state remains the same as in the previous time step. Note that the internal controller can also handle minimum operation time constraints; however, these are not shown in Figure 2 for the sae of simplicity. The control logic of algorithm A1 is shown in Figure 3. Denote by P PV the available PV power and by P l the total uncontrollable load. Similarly to A0, whenever any of the set-points T s7,min, T s10,max, T b,min, and T b,max is reached the HP turns on or off based on its internal controller. In any other case, the HP state is decided based on the net power P n, which is simply the power surplus in the building. In Figure 3, P HP,exp denotes the power that the HP is expected to consume if it turns on at the current time step. The parameter α є [0,1] determines which part of P HP,exp must be covered by P PV to allow HP operation. Higher α values will result in less frequent HP operation. Figure 3: Flowchart of algorithm A1. Figure 2: Flowchart of base case (algorithm A0) Figures 4-6 present the system operation under algorithm A2. The thermal storage is not utilized for PV energy management in this case; therefore, the HP operates based on A0. For this reason, to calculate P n the HP is treated as uncontrollable load. If power surplus (deficit) occurs and the battery is not fully charged (discharged), the charging (discharging) mode is enabled. If the KiBaM discharging or charging power constraints (8), (9) are active, any additional power is imported from or fed to the grid, respectively. If the SOC is 100% or below SOC min, then the battery idles. Algorithm 3 is described in Figure 7. Since the HP has priority over battery, the first part of A3 is identical to A1. After the HP state has been fixed, the battery is connected in charging or discharging mode depending on

5 P n. The same logic as before applies here for battery charging and discharging. Algorithms A1-A4, as well as the base case A0, can be expressed as a set of logic statements, which can be directly incorporated into Polysun via a component called programmable controller. This component allows logic statements to be programed in a high-level user friendly language. Also, specific operation intervals for thermal loads, or intervals during which operation is not allowed, can be defined using the same component. Figure 4: Flowchart of algorithm A2 Figure 6: Flowchart of battery control during discharging Figure 5: Flowchart of battery control during charging Algorithm 4, which is the most involved, is shown in Figure 8. The discharging phase is identical to A3; however, the charging priority is reversed. In presence of a power surplus, the battery is charged first. If the charging power constraint (9) is active or the battery is fully charged, the remaining power is consumed by the HP, provided that it is more than expected power consumption of the HP; otherwise, it is fed to the grid. Figure 7: Flowchart of algorithm A3

6 each, which leads to a total installed power of 10.8 Wp; a battery module Hoppece 24 OpzS 3000 with nominal cell voltage 2 V and a total capacity of 6 Wh. In this paper, we adopt the battery lifetime parameter values from [20]: α 1 = , α 2 = , α 3 = 4.90, α 4 = , and α 5 = In addition, the hot water temperature set-point is assumed 50 o C, and the average hot water demand 200 l/day. The uncontrollable electrical load (UL) was modeled adopting a profile with an annual consumption of 3103 Wh. The profile was generated using the load profile generator proposed in [22], which is available for download [23]. The profile generator simulates the behavior of the occupants based on a desire model, and includes typical operation patterns for more than 100 electrical devices. First, load curves are generated for each device using the desire model. Then, the total UL profile is calculated by adding up the energy use of each device at each point in time. The building diagram in Polysun with all components and details of the hydronic system is shown in Figure 9. The programmable controller is in the center of the diagram; its inputs are indicated with blue arrows, whereas the components that it controls with red arrows. Note that apart from the programmable controller, there two more components with the same icon. These are controllers for the internal electric heating element of the water tan and the position of the two-way mixing valves. We eep separate controllers for these devices, since they are not used for PV self-consumption maximization. The control parameters for algorithms A0- A4 are shown in Table I. Figure 9: The building diagram in Polysun Figure 8: Flowchart of algorithm A4 4 CASE STUDY In this paper, a single-family (2 parents both woring and 1 child) normal residential building is considered as a case study to show the algorithms performance. The following components are used: an HP with 10.1 W thermal power and 3.3 W electric power at W2/W35 operating conditions; a 500 l hot water storage tan; a rooftop PV system consisting of 60 panels rated at 180 W Table I: Controller parameters Tan temperature set-point T s7,min ( o C) 45 Tan temperature set-point T s10,max ( o C) 70 Building temperature set-point T b,min ( o C) 21 Building temperature set-point T b,max ( o C) 25 HP temperature set-point T HP,max ( o C) 65 Minimum battery charge level SOC min (%) 30 Parameter α 1 5 SIMULATIONS RESULTS In this section, we present annual simulation results for the base case and algorithms A1-A4 using Polysun. We also show the detailed system operation for two consecutive days in winter, spring, summer, and autumn.

7 Note that in these simulations the battery capacity is not optimized; this will be investigated in Section Annual results Tables II-VI present annual results for energy imports (exports) from (to) the grid, HP and total electricity consumption, battery charging and discharging energy (for algorithms A2, A3, and A4), PV self-consumption ratio, and the resulting electricity bill. Table II: Simulation results for Base Case (A0) Imports from grid (Wh) Exports to grid (Wh) HP consumption (Wh) Total consumption (Wh) PV self-consumption ratio (%) Electricity bill ( ) Table III: Simulation results for algorithm A1 Imports from grid (Wh) Exports to grid (Wh) HP consumption (Wh) Total consumption (Wh) PV self-consumption ratio (%) Electricity bill ( ) Table IV: Simulation results for algorithm A2 Imports from grid (Wh) Exports to grid (Wh) HP consumption (Wh) Total consumption (Wh) Battery charging (Wh) Battery discharging (Wh) PV self-consumption ratio (%) Electricity bill ( ) Table V: Simulation results for algorithm A3 Imports from grid (Wh) Exports to grid (Wh) HP consumption (Wh) Total consumption (Wh) Battery charging (Wh) Battery discharging (Wh) PV self-consumption ratio (%) Electricity bill ( ) Table VI: Simulation results for algorithm A4 Imports from grid (Wh) Exports to grid (Wh) HP consumption (Wh) Total consumption (Wh) Battery charging (Wh) Battery discharging (Wh) PV self-consumption ratio (%) Electricity bill ( ) Using only the HP and the building thermal inertia in A1 increases the PV self-consumption by approximately 1.5% compared to A0. On the other hand, the battery itself (A2) demonstrates a much higher potential and leads to PV self-consumption increase of roughly 15.5%. When both HP and battery are available, marginally higher self-consumption ratios are achieved when the priority during charging is given to the HP. Algorithm A3 leads to the highest PV self-consumption ratio, which is equal to 36.46%, i.e. more than one third of the available PV energy is consumed within the building premises. A3 gives the best results also in terms of savings for building owners, which amount to roughly 85 per year. Generally, increasing the PV self-consumption reduces the electricity bill. However, this is not the case for A1. Although A1 self-consumes more PV energy than A0, it results in marginally higher costs. This is because A1 operates the system in a less energy efficient way, increases the thermal losses, and eventually requires more electric energy for the HP. Usually this energy is imported from the grid, which brings additional costs. Clearly, the battery alone achieves most of the existing potential for PV self-consumption maximization. Comparing Tables IV and V, one can see that introducing also the HP in the control scheme further reduces the annual cost only by 6. However, this reduction comes at virtually zero cost, since the marginal cost of integrating the HP in the already installed controller is negligible. Similarly to A1, shifting the HP demand in A3 increases thermal losses and the annual HP consumption. Nevertheless, these additional losses are covered by battery discharge, rather energy imports from the grid. This explains why A3 achieves a lower cost than A2, whereas A1 increases the cost compared to A Daily operation results Figures show operation results from typical days in winter, spring, summer and autumn for A3, which demonstrates the best performance over the year. We focus on PV production, UL and HP consumption, battery charge and discharge energy, building temperature, battery SOC, and HP internal temperature. Figure 10: Production and consumption of system components during two typical days in winter (January) Figure 11: Building temperature, HP temperature, and

8 battery SOC during two typical days in winter (January) according to constraint (9). Similar observations can be Figure 12: Production and consumption of system components during two typical days in spring (April) Figure 14: Production and consumption of system components during two typical days in summer (July) Figure 13: Building temperature, HP temperature, and battery SOC during two typical days in spring (April) The seasonal patterns of HP operation can be clearly seen in the figures. The HP runs most of the time during a cold winter day, whereas it turns on only a few times during a summer day. In winter when PV production is low, HP operation is determined by the building heat demand, i.e. the HP consumes the minimum amount of energy that is required to eep building temperature right above 21 o C. Thus, the potential for PV energy management via HP control is very limited. In summer, the PV production is high; however, only a small part of the PV energy surplus is stored as thermal energy. This is because prolonged HP operation leads to overheating due to the low building heat demand. This can be seen in Figure 15, where the HP temperature remains often at or above T HP,max. This is in contrast to Figure 11, where the HP temperature in winter is significantly lower on average. Our simulations indicate that the potential for PV self-consumption maximization via HP control is higher in spring, and to a lesser extend in autumn. For example, the HP can effectively absorb the excess PV power during the first day of Figure 12, because high PV production coincides with increased heat demand and low HP temperatures (see Figure 13). If PV power surpluses exist during day-time, the battery exhibits the expected pattern: it charges during day-time and discharges in the evening and night hours to cover the load. Note that charging cycles might be interrupted by smaller discharging cycles in case the HP turns on and the PV power is not sufficient to provide the required power. In all cases, the rate of SOC increase reduces when the battery is nearly fully charged, Figure 15: Building temperature, HP temperature, and battery SOC during two typical days in summer (July) Figure 16: Production and consumption of system components during two typical days in autumn (October) Figure 17: Building temperature, HP temperature, and

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