- Email: [email protected]
Optimal design of energy conversion units for residential buildings considering German market conditions
Accepted Manuscript Optimal design of energy conversion units for residential buildings considering German market conditions Thomas Schütz, Markus Schraven, Sebastian Remy, Julia Granacher, Dominik Kemetmüller, Marcus Fuchs, Dirk Müller PII:
S0360-5442(17)31400-7
DOI:
10.1016/j.energy.2017.08.024
Reference:
EGY 11393
To appear in:
Energy
Received Date: 23 December 2016 Revised Date:
3 July 2017
Accepted Date: 6 August 2017
Please cite this article as: Schütz T, Schraven M, Remy S, Granacher J, Kemetmüller D, Fuchs M, Müller D, Optimal design of energy conversion units for residential buildings considering German market conditions, Energy (2017), doi: 10.1016/j.energy.2017.08.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
RI PT
Optimal design of energy conversion units for residential buildings considering German market conditions Thomas Sch¨ utz∗, Markus Schraven, Sebastian Remy, Julia Granacher, Dominik Kemetm¨ uller, Marcus Fuchs, Dirk M¨ uller
SC
RWTH Aachen University, E.ON Energy Research Center, Institute for Energy Efficient Buildings and Indoor Climate, Mathieustr. 10, Aachen, Germany
Abstract
M AN U
Many countries have passed governmental action plans to support the installation of renewable energy sources. However, most studies dealing with the optimization of building energy systems neglect a precise modeling of such subsidies, although these subsidies are specifically designed to strongly influence system setups. Therefore, this paper extends a model for the optimization of energy systems by a more accurate consideration of storage units and enhance
D
both models by accounting for major German pieces of legislation aimed at supporting renewable energies. Additionally, we consider typical German mar-
TE
ket characteristics, in particular the availability of multiple gas and electricity tariffs.
EP
We compare our model with the original formulation regarding a pure cost minimization and a forced reduction of CO2 emissions for three new buildings located in Germany. The results imply that the considered subsidies strongly
AC C
support the installation of PV modules and CHP units. Without these subsidies, batteries and solar thermal collectors become more important. Additionally, the findings illustrate that the new storage model is slightly more accurate, but only marginally affects the total annual costs and required computing times. The conducted sensitivity analysis has shown that the obtained results are relatively ∗ Corresponding
author Email address: [email protected] (Thomas Sch¨ utz)
Preprint submitted to Energy
August 7, 2017
ACCEPTED MANUSCRIPT
robust to variations in energy tariff costs and demands. Keywords: Building Energy Systems, German regulations, Mixed-Integer
1
RI PT
Linear Programming, Multiple energy tariffs
1. Introduction
The transition towards a more energy efficient economy with lower CO2
3
emissions is a recognized objective of the European Union [1]. In Germany, this
4
concept is known as “Energiewende” and aims at reducing greenhouse gas emis-
5
sions, increasing electricity generation from Renewable Energy Sources (RES)
6
and achieving higher energy efficiency in general [2]. In the context of buildings,
7
which account for approx. 40% of total energy consumption in the European
8
Union [1], emission reductions and energy savings can for example be achieved
9
by installing more efficient heating devices and by improving their control strat-
M AN U
10
SC
2
egy.
In recent years, many different heat and electricity generation as well as
12
storage technologies evolved for application in buildings. Small-scale Combined
13
Heat and Power (CHP) units offer a highly efficient method for generating heat
14
and electricity simultaneously from fossil fuels [3, 4]. Potential benefits can
15
further be leveraged by introduction of Thermal Energy Storage (TES) devices
16
[5]. Heat Pump (HP) systems present a technology for efficiently using electricity
17
for heating purposes [6]. RES, especially solar systems can further be used on
18
building level, such as Solar Thermal Collectors (STCs) [7] or Photovoltaic (PV)
19
modules [8]. The integration of such fluctuating generators can for example be
21
22
23
TE
EP
AC C
20
D
11
enhanced by storage devices such as TES [9] and batteries (BATs) [10]. In order to achieve the proposed emission reductions and energy savings, the
German government supports the utilization of technologies such as RES and CHP in the building sector. For example, the Renewable Energy Sources Act
24
(German abbreviation EEG) [11], guarantees above market and long-term feed-
25
in tariffs for PV plants. Additionally, the Act on Combined Heat and Power
26
Generation (German abbreviation KWKG) [12] provides subsidies for feed-in
2
ACCEPTED MANUSCRIPT
and self-consumed electricity from CHPs. Further methods for promoting RES
28
and low CO2 technologies include subsidies from the German Reconstruction
29
Credit Institute (German abbreviation KfW) for combined PV and BAT systems
30
[13], tax exemptions for CHP units [14] as well as private utility companies
31
offering reduced electricity tariffs for HP systems.
32
1.1. Literature review
RI PT
27
In order to evaluate the economic suitability of a small number of predefined
34
Building Energy Systems (BESs), multiple, simplified simulation models have
35
been developed [15, 16, 17].
M AN U
SC
33
However, the vast quantity of combinations of available devices into a BES
37
and the specific subsidies require a systematic analysis and evaluation method
38
[18], making optimization approaches a viable option for determining the op-
39
timal structure, sizing and operation of BES. There are already optimization
40
frameworks available for energy system optimization, such as TIMES [19], DER-
41
CAM [20], and COMPOSE [21, 22]. These models may contain shortcom-
42
ings, such as oversimplifications leading to effects like the technology mix effect
43
[23] that can hardly be altered in such given frameworks. Therefore, many
44
researchers develop their own models for optimizing BES design, sizing, and
45
operation.
TE
D
36
The inherent nonlinearities arising in such optimization models, like the
47
typically nonlinear part load behavior of generation units, has led to the devel-
48
opment of accurate mixed-integer nonlinear programs (MINLP) [24, 25]. Ac-
49
cording to Klatt and Marquardt [26], however, MINLP is still not a suitable
51
52
53
AC C
50
EP
46
method for reasonably sized models. Therefore, MINLP models are often reformulated and solved using mixed-integer linear programming (MILP) or handled with heuristics [27, 28]. However, using heuristics has proven to be very time consuming and does not necessarily lead to significant improvements regarding
54
the obtained accuracy, if the model can be linearized properly [29]. Addition-
55
ally, MILP models can further be used as a good initial solution for a MINLP
56
solver [30]. 3
ACCEPTED MANUSCRIPT
Therefore, the majority of analyses dealing with the optimal design and
58
operation of BES use MILP, often requiring major simplifications. These sim-
59
plifications most frequently occur regarding the devices’ capacities and their
60
part load behavior. Device capacity can either be modeled in a continuous
61
manner or discretely, as illustrated in Figure C.1 for all CHP units considered
62
in this paper. Discrete modeling allows for assigning specific investment costs,
63
efficiency curves, nominal heat outputs, operating times, etc. to each device
64
individually, whereas in a continuous device modeling, a representative device
65
is scaled between lower and upper limits for the nominal heat or electricity out-
66
put. In the continuous model, all part load efficiency curves are the same for
67
one type of device (e.g. for all CHP units) and costs are typically based on lin-
68
ear regressions, as shown in Figure C.1. Additionally, the continuous approach
69
can lead to optimal results that are not available for purchase (e.g. the calcula-
70
tion requires a 15 kW CHP unit, whereas only 10 kW and 20 kW are available).
71
Therefore, the continuous approach presents a major simplification that reduces
72
the model’s accuracy, while typically improving the computing times.
M AN U
SC
RI PT
57
Furthermore, multiple approaches for modeling part load are available in
74
MILP models. Figure C.2 shows three different models for the electrical effi-
75
ciency of one CHP unit considered in this paper. The dashed blue line depicts
76
a model that neglects the switching on threshold and does not account for part
77
load deterioration. The red line shows a model that also does not consider part
78
load deterioration but accounts for a switching on threshold of the device. In
79
contrast, the black curve displays a piecewise linearization of the device’s part
80
load behavior, which has for example been considered by Pickering et al. [29].
82
83
84
TE
EP
AC C
81
D
73
The following paragraphs present studies dealing with the optimal design, sizing, and operation of energy systems using MILP and illustrate their underlying simplifications.
Ashouri et al. [31] presented a MILP framework for the optimal selection and
85
sizing of smart building systems in Switzerland, considering all aforementioned
86
generation and storage units, as well as chillers and ice storage systems. Their
87
device modeling considers continuous equipment sizes rather than available, 4
ACCEPTED MANUSCRIPT
discrete sizes. Furthermore, no switching on threshold or decreased part load
89
performance are modeled. Merkel et al. [32] used the same simplifications,
90
optimizing residential micro-CHP systems in the United Kingdom, considering
91
peak load boilers and hot water storage tanks.
RI PT
88
Other studies [33, 34] use this continuous dimensioning as done by Ashouri et
93
al. [31], however accounting for linear part load losses. Ameri and Besharati [33]
94
optimize district heating and cooling networks in Iran considering gas turbines,
95
boilers, chillers and PV. Voll et al. [34] compute an optimal energy system for
96
industrial applications in Germany comprising CHPs, boilers and chillers.
SC
92
Mehleri et al. [35, 36] model some equipment sizes, such as boilers, PV area
98
and TES volume with continuous variables, whereas CHP capacity is described
99
with discrete steps. They also neither account for switching on thresholds nor
100
part load deterioration. Their model is applied to optimize local neighborhoods
101
in Greece considering CHPs, boilers, PV, TES and district heating as well as
102
microgrids. They account for market characteristics by considering different
103
constant feed-in tariffs for electricity from CHP and PV.
M AN U
97
Such a mixed modeling of device capacities is also used by [37, 38]. Harb
105
et al. [37] model CHP and HP capacities discretely and rely on the continuous
106
sizing for PV, boilers and TES. In this model, CHP part load is handled with
107
an empirical approach, whereas constant efficiencies are assumed for boilers and
108
HP units for the entire modulation range. The model is applied to determine
109
optimal configurations for German residential buildings and extended to com-
110
pute local heating networks and microgrids for a small neighborhood. Harb
111
et al. [37] consider some aspects of German regulations, such as above-market
113
114
115
TE
EP
AC C
112
D
104
feed-in remuneration for electricity generated through PV as well as support for CHP units like tax exemptions for gas combusted in CHP units. However, they did not consider regulations impeding the operation and installation of specific energy conversion components, such as the EEG levy on self-consumption if the
116
installed capacity exceeds a certain threshold, a limit on the maximum feed-in of
117
PV as well as capacity specific remuneration for PV. Additionally, to the best of
118
our knowledge, this approach presents the only available model for considering 5
ACCEPTED MANUSCRIPT
two electricity tariffs during the design optimization, a special heat pump tariff
120
and a standard electricity tariff. Lozano et al. [38] present the optimization
121
of combined heat, cooling and power systems considering CHP, boilers, chillers
122
and TES for a city district in Spain. They model device capacities discretely
123
but represent TES sizes continuously. Their devices are assumed to be 2-point
124
controlled, not modeling part load. Furthermore, Lozano et al. [38] investigate
125
the effect of formerly required minimum self-consumption rates of electricity
126
generated through CHP units on the design and sizing of CCHP systems.
SC
RI PT
119
Buoro et al. [39] analyze fully automated homes in Italy by optimizing their
128
respective energy systems consisting of CHP, boiler, chiller, PV, STC and TES.
129
The part load is based on a linear regression without accounting for the devices’
130
activation threshold.
M AN U
127
Renaldi et al. [40] describe a framework for optimizing HP and TES sys-
132
tems for residential buildings in the United Kingdom. Their devices’ selection is
133
entirely based on discrete choices, however their TES model assumes constant
134
losses, even if the storage is totally discharged. Heat pumps’ temperature-
135
dependent COP is modeled, however the COP is assumed to be constant during
136
part load. Renaldi et al. [40] consider a governmental support for heat gener-
137
ation from HP with a linear relation between this revenue and the building’s
138
annual heat demand.
TE
D
131
Wakui and Yokoyama [41] present a model for optimizing energy systems
140
for Japanese residential buildings comprising CHP, TES, boiler and electrical
141
heaters (EHs). The selection of CHP and storage tanks is coupled, so that if
142
a certain CHP unit is chosen, a pre-specified storage tank is installed as well.
144
145
146
147
AC C
143
EP
139
Part load behavior is described by means of piecewise linearization, introducing multiple linear relationships that model the nonlinear part load curves. Wakui et al. [42] extended their model by further considering heat pumps. The selection of TES units has been decoupled from the selection of CHP units in this work; however, TES’ sizes are chosen via continuous variables rather than discretely.
148
In conclusion, the model developed by Wakui et al. [42] overcomes most
149
of the described simplifications applied in MILP models. We therefore used 6
ACCEPTED MANUSCRIPT
this model as a foundation for this work. Additionally, the literature review
151
shows that modeling of subsidies and market characteristics has largely been
152
overlooked or treated in a simplified manner in previous works. Remuneration
153
for electricity or heat has commonly been coupled linearly to the generation,
154
for instance neglecting decreased remuneration for large facilities. Additionally,
155
regulations impeding the operation of specific devices, such as feed-in limits or
156
costs for self-consumption have not been taken into account thoroughly.
157
1.2. Contributions
SC
RI PT
150
In order to contribute to the field of optimal design of building energy sys-
159
tems, we extend the model proposed by Wakui et al. [42] regarding device se-
160
lection and the implementation of German governmental subsidies and market
161
characteristics. Our work therefore presents the following three novelties.
M AN U
158
First, we extend the original model formulation to also select storage units
163
discretely. In this way, we assure that the determined, optimal system is avail-
164
able for purchase in reality, and we are able to include detailed information such
165
as standby losses, charging and discharging characteristics as well as investment
166
costs for each type of storage unit. Such discrete modeling has partly been done
167
by the studies cited above; however, to the best of our knowledge, TES have
168
not been accurately described in a discrete manner.
TE
D
162
The second novelty is a detailed modeling of many specific German regula-
170
tions and market characteristics, of which some support and others impede the
171
installation and operation of RES. Based on the current EEG, we account for
172
limitations on PV feed-in. We consider variable feed-in remuneration for PV,
174
175
176
AC C
173
EP
169
depending on the installed capacity as well as the EEG levy for self-consumed electricity if the installed peak generation capacity exceeds 10 kW. This paper models the KWKG considering remuneration for feed-in and self-consumption for electricity generated with CHP. Furthermore, the presented approach models
177
tax exemptions for CHP units’ fuel. Also, we consider subsidies for PV systems
178
with batteries according to KfW 275. Additionally, we account for reduced heat
179
pump tariffs that are often offered by utility companies in Germany. Since most 7
ACCEPTED MANUSCRIPT
of the previously mentioned papers use generic settings, the consideration of
181
local market characteristics improves the applicability of the model for realistic
182
investment decisions. In order to facilitate the transfer of this paper’s findings
183
to incentive programs and market characteristics in other countries, we present
184
the modeling approach in general formulations and detail the distinct features
185
of the German application.
RI PT
180
Our third contribution is the detailed modeling of multiple gas and electricity
187
tariffs. Previous studies only accounted for a single gas and electricity rate, or
188
in the case of Harb et at. [37] a single gas tariff and two electricity tariffs,
189
whereas we include the possibility of considering an arbitrary number of gas
190
and electricity tariffs and we also account for a tiered pricing structure of each
191
tariff. In this way, we allow for analyzing the trade-off between monetary costs
192
and environmental impacts.
M AN U
SC
186
These novelties are of manifold importance: Researchers benefit from a de-
194
tailed decision modeling. Regulators can investigate the effect of certain laws
195
on rational decisions. Practitioners can use the framework for determining op-
196
timal BES configurations and analyze trade-offs between economic and ecologic
197
objectives.
TE
D
193
The rest of this paper is structured as follows: Section 2 describes the devel-
199
oped optimization model. Afterwards, Section 3 presents the inputs for analyz-
200
ing our model. In order to evaluate our model extensions, we first compare our
201
developed model with the original formulation in Section 4. This comparison
202
is conducted for three new German residential buildings significantly differing
203
in size and location. Furthermore, we evaluate the importance of considering
205
206
207
208
AC C
204
EP
198
governmental subsidies and market characteristics on BES design, sizing and operation as well as the estimation of total costs. We also perform a sensitivity analysis for analyzing the robustness of the calculated solutions and discuss the weaknesses of our model. Finally, the findings are summarized and an outlook for future research is given in Section 5.
209
The model and calculations described in this paper can be downloaded
210
from https://github.com/RWTH-EBC/BESopt. (The currently private repos8
ACCEPTED MANUSCRIPT
itory will be made public upon acceptance of this paper.)
212
2. Modeling
RI PT
211
The structure of the considered building energy system is shown in Fig-
214
ure C.3. Conventional heat generators such as gas boilers, CHP units, EHs,
215
continuous flow water heaters (CFWHs) and electrical air/water HPs are con-
216
sidered. Furthermore, storage devices like BATs and TES units are available.
217
Solar generators like PV modules and STCs as well as peripheral devices like
218
inverters (INVs) are also included as possible parts of the optimal BES. Ther-
219
mal loads include space heating (SH) as well as domestic hot water (DHW).
220
Electrical loads describe the building’s electricity consumption for lighting and
221
electrical appliances like computers, refrigerators and televisions that we cur-
222
rently consider unshiftable. The implemented energy balances are described in
223
this chapter and summarized graphically in Appendix A.
M AN U
SC
213
The model requires annual inputs for thermal and electrical loads. Further
225
inputs include device-specific data, such as the available types (e.g. CHP type 1,
226
CHP type 2, etc.) of each device (e.g. CHP, HP, etc.) and their characteristics,
227
for instance efficiency curves, expected life time, investment costs as well as
228
operation and maintenance costs. Also, information regarding different gas and
229
electricity tariffs, economic parameters like interest rate, tax rates, length of the
230
observation period and subsidy rates can be specified.
EP
TE
D
224
Since modeling an entire year requires long computing times, the inputs are
232
clustered into multiple representative periods that are weighted with weighting
233
234
235
236
AC C
231
variables wd . The clustering is based on the k-medoids method [43] and additionally rescales the cumulated inputs in order to preserve the annual energy demands [44]. This method has been shown to provide reliable results of high quality for energy system optimization purposes [45].
237
A main novelty of this paper is the consideration of many German subsidies
238
and market characteristics. Table C.4 summarizes the implemented character-
239
istics and links them to the corresponding equations used in this paper.
9
ACCEPTED MANUSCRIPT
240
The remainder of this section explains the objective function as well as the economic, technical and ecological modeling.
242
2.1. Objective function
RI PT
241
The original model of Wakui et al. [42] minimizes the annual primary energy
244
consumption. However, we believe that most decisions regarding energy supply
245
are economically driven. Thus, we chose to minimize annual costs cann . An-
246
nual costs are the sum of costs for investments cinv , operation and maintenance
247
co&m , demand related costs cdem , metering cmet and EEG-levy for self-consumed
248
electricity ceeg less the revenues generated from feed-ins efeed and subsidies esub .
M AN U
SC
243
min cann = cinv + co&m + cdem + cmet + ceeg − efeed − esub
(1)
The following subsections focus on the constraints that are new in this pa-
250
per. Additionally required constraints are briefly described and listed in Ap-
251
pendix B as well as this project’s open-source repository (https://github.
252
com/RWTH-EBC/BESopt).
253
2.2. Economic constraints
The economic modeling is based on the German engineering guideline VDI 2067
TE
254
D
249
[46], that has also been used in the authors’ previous publications [37, 45, 47, 48].
256
Investment costs are distributed into equal, annual payments by means of
257
the capital recovery factor CRF . For STC and PV, the investment costs are
258
determined by the number of modules and their specific costs, whereas for other
259
devices the unit’s costs can be used directly.
261
262
263
AC C
260
EP
255
Costs for operation and maintenance for all devices but CHP units are mod-
eled as a fixed percentage of the initial investment. The corresponding values can be found in [46]1 . For CHP units, ASUE [49] offers more detailed models
for operation and maintenance related costs that are based on device operation, 1 These
costs are based on a German engineering guideline that is currently in force and
are therefore considered to be suitable. However, the framework allows for defining accurate values for each device, if such data are available.
10
ACCEPTED MANUSCRIPT
assuming a proportionality between the generated amount of electricity and the
265
resulting costs.
266
2.2.1. Demand related costs
RI PT
264
The modeling of demand related costs is a novelty of this publication, since
268
an arbitrary number of gas and electricity tariffs can be considered. In contrast,
269
all cited previous publications only used one gas tariff and at most two electricity
270
tariffs.
SC
267
Demand related costs depend on the chosen gas or electricity tariff. If a CHP or boiler is installed, the binary decision variable xdev,i indicating if type i
M AN U
of device dev has been purchased, would equal 1. In this case, a gas tariff tarjgas has to be selected. Furthermore, at most one gas tariff can be chosen. X X tarjgas ≥ xdev,i ∀ dev ∈ {CHP, BOI} j
X
i
tarjgas ≤ 1
j
(2) (3)
As many utilities offer a tiered pricing structure depending on the annual
272
gas,lvl consumption, different tariff levels tarj,l are introduced. At most one level
273
can be valid and if the corresponding gas tariff is not chosen, all level variables
274
have to be 0:
TE
D
271
X
gas,lvl tarj,l = tarjgas
∀j
(4)
l
The correct level is bounded by a lower bound on the minimum annual
276
LB consumption Gtarj,l and an upper bound on the maximum annual consumption
277
UB Gtarj,l :
AC C
EP
275
gas,lvl gas,lvl LB CHP UB tarj,l · Gtarj,l ≤ GBOI ≤ tarj,l · Gtarj,l j,l + Gj,l
278
279
280
281
∀j, l
(5)
In this equation, GBOI and GCHP describe the annual gas consumption of j,l j,l
CHP units and boilers at tariff j and level l. For boilers and CHP units, the annual gas consumption Gann dev of all days d
and time steps t results in: X XX Gann wd · ∆t · E˙ dev,i,d,t dev = d
t
i
11
∀dev ∈ {BOI, CHP }
(6)
ACCEPTED MANUSCRIPT
In this equation, wd stands for the weighting variable for typical demand day
283
d. Since this day represents wd other days of the original input data set, the gas
284
demand at every time is weighted appropriately. Furthermore, ∆t denotes the
285
length of each time step, which has been set to one hour in this work. However,
286
if more detailed inputs are available, the temporal resolution can be adjusted
287
accordingly.
tariff: Gann dev =
XX j
290
SC
289
The annual gas consumption is distributed among the different levels of each
Gdev j,l
∀dev ∈ {BOI, CHP }
l
gas cdem · CRF · BOI = b
XX j
291
292
(7)
The demand related costs for gas consumption of boilers are computed with:
M AN U
288
RI PT
282
var,gas GBOI j,l · cj,l
(8)
l
According to the German Energy Tax Act, a tax refund of taxgas that reduces the variable costs of cvar,gas can be obtained for CHP units [14]: j,l XX gas cdem · CRF · GCHP · cvar,gas − taxgas CHP = b j,l j,l
(9)
l
D
j
Demand related electricity costs for purchases from the grid are modeled
294
similar to the costs for natural gas. In contrast to the gas tariffs, two electricity
295
tariffs may be selected from the available set of tariffs. One electricity tariff has
296
to be chosen to satisfy plug loads:
298
299
300
EP
X
tarjel∗ = 1
(10)
j∗
In Equation 10, j ∗ denotes all electricity tariffs that can be chosen for plug
AC C
297
TE
293
loads and are not special HP tariffs. If a HP unit is installed, a special HP tariff hpt, valid for electricity required
for the HP, CFWH and EH, may be selected: X
el tarhpt ≤
X
xHP i
(11)
i
hpt 301
For all types of tariffs, a tiered pricing structure is considered. These tiered
302
pricing structures for electricity tariffs are modeled in a similar manner as for 12
ACCEPTED MANUSCRIPT
gas tariffs. At most one level can be valid and if the corresponding electricity
304
tariff is not chosen, all level variables have to be 0: X
el,lvl tarj,l = tarjel
∀j
(12)
l 305
306
RI PT
303
LB The correct level is bounded by a minimum annual consumption Etarj,l and UB a maximum annual consumption Etarj,l :
∀j, l
SC
el,lvl el,lvl LB HP PL UB tarj,l · Etarj,l ≤ Elj,l + Elj,l ≤ tarj,l · Etarj,l
(13)
HP PL In this equation, Elj,l and Elj,l describe the annual electricity purchases
308
for HP and the house’s plug loads at tariff j and level l. Since the HP tariff
309
PL cannot be used to cover the house’s plug loads, Elhpt,l is set to 0.
310
The annual electricity imports for types HP and PL are: ann Eltype =
X
wd · ∆t ·
X t
d
312
levels of each tariff:
type Elj,l
317
(15)
Finally, the demand related costs for electricity are computed with:
EP
XXX type
j
type Elj,l · cvar,el j,l
(16)
l
AC C
2.2.2. Metering costs
Metering costs directly follow from the chosen tariff and the resulting level: cmet type =
316
∀type ∈ {HP, P L}
l
cdem = bel · CRF · el
315
(14)
TE
XX j
314
∀type ∈ {HP, P L}
The annual amount of electricity imports is distributed among the different
ann Eltype =
313
type,imp Pd,t
D
311
M AN U
307
XX j
type,lvl tarj,l · cfix,type j,l
∀type ∈ {el, gas}
(17)
l
In this equation, cfix,type denotes the fixed, annual metering costs of tariff j j,l
and level l.
13
ACCEPTED MANUSCRIPT
318
2.2.3. EEG levy In Germany, end-customers pay the so-called EEG levy to support RES.
320
According to the EEG [11], a certain percentage of the EEG levy has to be
321
paid to the transmission system operator for self-consumed electricity, if the
322
installed capacity exceeds 10 kW. Equations 18 and 19 describe whether or not
323
more than 10 kW capacity of CHP or PV are installed. In these equations, xeeg
324
denotes a binary variable that is equal to 1 if more than 10 kW are installed and
325
Meeg,cap is a big-M, serving as an artificial upper bound for installed generation
326
capacity, set to 1000 kW.
M AN U
X
SC
RI PT
319
nom PPV,i · zPV,i ≤ 10 + xeeg · Meeg,cap
(18)
nom PCHP,i · xCHP,i ≤ 10 + xeeg · Meeg,cap
(19)
i
X i
eeg The following two equations describe the resulting costs c˜dev if this 10 kW
threshold is not enforced: X
wd ·
X
D
eeg c˜PV = f eeg · ∆t ·
d
X
wd ·
TE
eeg c˜CHP = f eeg · ∆t ·
d
PL HP PPV,d,t + PPV,d,t
(20)
t
XX t
PL HP PCHP,i,d,t + PCHP,i,d,t
(21)
i
PL HP In these equations, Pdev and Pdev describe the portions of self-generated elec-
328
tricity that are self-consumed by the house’s plug loads and HP operation and
329
are thus not fed into the grid. Additionally, f eeg = 0.02752 Euro/kWh is the
330
cost for self-consumed electricity in 2017, which is currently equal to 40% of the
331
total EEG levy [11, 50].
AC C
EP
327
The following three equations represent the linearized formulation of the
eeg product of c˜dev · xeeg [51] and therefore describe the billed EEG levy under
consideration of the 10 kW threshold: eeg cdev ≤ xeeg · Meeg,billed
∀dev ∈ {CHP ; P V }
(22)
eeg eeg c˜dev − cdev ≥0
∀dev ∈ {CHP ; P V }
(23)
eeg eeg c˜dev − cdev ≤ (1 − xeeg ) · Meeg,billed
∀dev ∈ {CHP ; P V }
(24)
14
ACCEPTED MANUSCRIPT
In this reformulation, Meeg,billed is an upper bound for the total amount of billed
333
EEG levy, which has been set to 1,000,000.00 Euro/a.
334
2.2.4. Revenues from feed-in
RI PT
332
Furthermore, according to the EEG [11], the remuneration rate for elec-
tricity from PV that is fed into the grid depends on the installed capacity. Below 10 kW, feed-in is remunerated with pfeed,PV,10 = 0.1264 Euro/kWh, be-
SC
tween 10 and 40 kW pfeed,PV,40 = 0.123 Euro/kWh and between 40 and 750 kW
pfeed,PV,750 = 0.1103 Euro/kWh [52]. The following equations are used for de-
X
M AN U
termining the correct interval:
nom PPV,i · zPV,i ≤ 10 · xeeg,10 + 40 · xeeg,40 + 750 · xeeg,750
i
X
xPV,i = xeeg,10 + xeeg,40 + xeeg,750
i
(25) (26)
335
The total amount of sold electricity is split into the corresponding categories,
336
sell,PV where El40 for instance describes the amount of electricity sold at pfeed,PV,40 .
X
337
X t
341
342
EP
∀size ∈ {10, 40, 750}
(28)
Finally, the feed-in remuneration for PV results in:
AC C
340
(27)
Additionally, at most one of these three variables can be unequal to zero:
X
EEX efeed · CRF · PV = b
339
sell,PV Elsize
size∈{10,40,750}
sell,PV Elsize ≤ MPV,sell · xeeg,size
338
X
sell PPV,d,t =
TE
d
wd ·
D
sell,PV Eltotal = ∆t ·
sell,PV pfeed,PV,size · Elsize
(29)
size∈{10,40,750}
Electricity surplus from CHP can be fed into the grid at standard mar-
ket rates. In this work, we use a time-independent average market rate of pfeed,CHP = 0.038 Euro/kWh. This value represents the average CHP index in 2016 [53] and avoided grid costs. EEX efeed · CRF · pfeed,CHP · ∆t · CHP = b
XX i
15
d
wd ·
X t
sell PCHP,i,d,t
(30)
ACCEPTED MANUSCRIPT
344
345
346
347
2.2.5. Subsidies In this paper, we consider subsidies for CHP units according to [12] and subsidies for BAT systems based on [13].
RI PT
343
For large micro CHP units above 2 kW rated electrical power (subset i∗ ), the subsidies are computed as:
i XX X h sub,CHP sub,CHP EEX PL HP sell esub ·CRF ·∆t· wd · pself · PCHP,i · PCHP,i ∗ ,d,t + PCHP,i∗ ,d,t + p ∗ ,d,t CHP,large = b sell
348
In this equation,
d
t
psub,CHP self
SC
i∗
(31)
= 0.04 Euro/kWh denotes the subsidies for self-
consumed electricity from CHP units and psub,CHP = 0.08 Euro/kWh the subsell
350
sidies for sold power [12].
351
352
M AN U
349
Smaller micro CHP units (subset i∗ ) can either receive a fixed or a variable subsidy. The fixed subsidies are computed with: subfix = CRF · tmax · psub,CHP · fix
X
nom xCHP,i∗ · PCHP,i ∗
(32)
i∗
354
Here, tmax stands for the maximum subsidized time period of 60,000 full load hours and psub,CHP = 0.04 Euro/kWh is the specific subsidy. fix
D
353
TE
Variable subsidies for small scale CHP units are computed just like for large scale CHPs as described in Equation 31, however they are stored in variable
AC C
ner [51].
EP
subvar . Since the maximum of both, subfix and subvar will be used by investors, fix var the following equations model esub in a linear manCHP,small = max sub ; sub
fix esub CHP,small ≥ sub
(33)
var esub CHP,small ≥ sub
(34)
fix esub + M fix · δ var CHP,small ≤ sub
(35)
var esub + M var · (1 − δ var ) CHP,small ≤ sub
(36)
355
In this set of equations, M fix = CRF · tmax · psub,CHP · 2 kW is an upper fix
356
bound for subfix , M var = bEEX · CRF · 8760 h · psub,CHP · 2 kW for subvar and sell
357
δ var denotes a binary variable that is 1 if the variable subsidy option is chosen.
16
ACCEPTED MANUSCRIPT
The overall subsidies for CHP units result in:
358
sub sub esub CHP = eCHP,small + eCHP,large
RI PT
(37)
Batteries are subsidized by the German Reconstruction Credit Institute (KfW) [13]. This subsidy is essentially a relatively cheap credit for the installation of BATs in combination with PV modules. Since only 81% of this
SC
credit have to be repaid, we consider the remaining 19% of this credit as a sub-
sidy for BAT storages [13]. The subsidies are capped by the minimum of two factors. These subsidies are limited by submax BAT = 2, 000.00 Euro/kW times the
M AN U
installed PV peak power and they also do not exceed the actual investments reduced by subBAT = 1, 600.00 Euro/kW times the installed PV peak power: max sub esub BAT ≤ 19% · CRF · subBAT · ζBAT
inv inv sub esub BAT ≤ 19% · cPV + cBAT − CRF · subBAT · ζBAT
(38)
(39)
sub In this set of equations, ζBAT stands for the product of installed PV power P nom P PPV,i · zPV,i and the decision if a battery storage is installed xBAT,i . i
D
i
According to Williams [51], this product is linearized as follows: X
nom PPV,i · zPV,i
(40)
TE
sub ζBAT ≤
i
sub ζBAT
≥
EP
X Amax sub nom ζBAT ≤ max PPV,j · · xBAT,i j APV,j i X
nom PPV,i
· zPV,i −
359
360
361
362
! 1−
X
xBAT,i
i
Amax nom · max PPV,j · j APV,j
(42)
AC C
i
(41)
2.3. Technical constraints This section describes the technical constraints, which model the device se-
lection and their operation. 2.3.1. Device selection
363
The nominal heat provided by CHP, EH, boiler and HP has to match or ex-
364
ceed the design heat load (DHL) to ensure thermal comfort during cold weather
17
ACCEPTED MANUSCRIPT
conditions: X
X dev∈{BOI,CHP,EH,HP }
366
˙ DHL xdev,i · Q˙ nom dev,i ≥ Q
(43)
RI PT
365
i
2.3.2. Generating devices
Similar to Wakui and Yokoyama [41] and Wakui et al. [42], we also use a
368
piecewise linearization for modeling the part load behavior of CHP units and
369
HP units but also apply this technique to boilers. The corresponding equations
370
are presented in Appendix B. The operation of CFWHs and EHs is simplified
371
as we assume a loss-free conversion from electricity to heat and no minimum
372
activation limits during part load operation.
M AN U
SC
367
373
For the heat generation of STC modules, an efficiency ηSTC,i,d,t is computed
374
that accounts for the optical efficiency η0 as well as linear k1 and quadratic k2
375
thermal losses [54]: ηSTC,i,d,t = η0 − k1 ·
376
difference ∆TST C describes the difference between the average collector fluid
377
amb temperature and ambient temperature Td,t . In this work, we assume a constant
378
STC flow temperature of 35◦ C. The solar irradiation onto the collector is Id,t
379
which is computed according to [54, 55].
− k2 ·
2 ∆TST C Id,t .
The temperature
D
∆TST C Id,t
The electricity generation from PV is modeled similarly, considering the aver-
381
age efficiency of inverters η¯INV . Since the nominal efficiencies of all inverters con-
382
sidered in this paper vary between 97% and 98%, an average efficiency of 97.7% is
383
used for all inverters. Furthermore, the module’s efficiency is based on Dubey et
386
387
388
389
EP
385
al. [56], considering the solar irradiation onto the module as well as temperature h i Id,t amb ref amb effects: ηPV,i,d,t = η0 · 1 − γ · Td,t − TPV,i + T NOCT − Td,t · I NOCT . In
AC C
384
TE
380
this equation, γ is the loss coefficient, T NOCT and I NOCT stand for the temref perature and irradiation at NOCT conditions (20◦ C, 0.8 kW/m2 ) and TPV,i
describes the cell temperature at these conditions. The power from PV, CHP and the power discharging the BAT is split into
390
self-consumption for general plug loads (PL), the HP and electricity fed into
391
the grid (sell). This splitting is necessary for assigning the right amounts of
392
electricity to each tariff. For BAT and PV, these parts can be combined for all
18
ACCEPTED MANUSCRIPT
types of batteries and modules: X
PL HP sell Pdev,i,d,t = Pdev,d,t + Pdev,d,t + Pdev,d,t
∀dev ∈ {BAT, P V } , d, t (44)
i 394
395
RI PT
393
In case CHP units are chosen, a strict distinction between each type of CHP electricity has to be made due to the precise modeling of the KWKG: PL HP sell PCHP,i,d,t = PCHP,i,d,t + PCHP,i,d,t + PCHP,i,d,t
∀i, d, t
(45)
According to the EEG [11], at most 70% of the PV peak power may be fed
397
into the distribution grid. If a battery system is installed and the subsidies described in Section 2.2.5 are chosen, this number is even reduced to 50% [13]. h i X sell nom PPV,d,t ≤ PPV,i 0.7 · zPV,i − ζiPV,BAT + 0.5 · ζiPV,BAT ∀d, t (46)
M AN U
398
SC
396
i
In Equation 46, ζiPV,BAT stands for the product of zPV,i and is linearized as follows [51]: ζiPV,BAT ≤ zPV,i A · APV,i
X
xBAT,j
D
ζiPV,BAT ≤
max
TE
ζiPV,BAT ≥ zPV,i − 1 −
X
xBAT,j ·
j
EP
nom zPV,i · PPV,i ≤
X
i
402
403
(47)
∀i
(48)
Amax APV,i
∀i
(49)
nom xINV,j · PINV,j ·
(50)
j
2.3.3. Storages
AC C
401
∀i
The inverter is sized according to the installed, nominal PV power: X
400
xBAT,j that
j
j
399
P
Storage units’ energy contents are modeled based on their state of charge
Sdev,i,d,t . Storages can only be charged (Sdev,i,d,t > 0) if the corresponding
device has been purchased: Sdev,i,d,t ≤ xdev,i
∀dev ∈ {BAT, T ES} i, d, t
(51)
404
In order to allow a discrete selection of storage units, an energy balance
405
is modeled for each type and each time step. In previous works that used a 19
ACCEPTED MANUSCRIPT
continuous device modeling for TES units, only one equation for each time
407
step is necessary, whereas the discrete device modeling increases the amount of
408
required ‘state of charge’-variables by the number of considered TES and BAT
409
units.
Sdev,i,d,t = (1 − ϕdev,i )·Sdev,i,d,t−1 +∆t·
ηdev,i · chdev,i,d,t − dchdev,i,d,t capdev,i
RI PT
406
∀dev ∈ {BAT, T ES} i, d, t (52)
The storage’s relative standby losses between two consecutive time steps are
411
ϕdev,i , ηdev,i describes the storage cycle’s efficiency, chdev,i,d,t and dchdev,i,d,t
412
stand for the charging and discharging power and capdev,i for the storage’s
413
capacity. The capacity can directly be derived from data sheets for BATs. For
414
TES units, we calculate capdev,i = ρ·κ·Vi ·∆T max , where ρ and κ are the density
415
and specific heat capacity of water, Vi is the storage’s volume and ∆T max = 35 K
416
the maximum temperature spread inside the tank.
M AN U
SC
410
Since we are using typical demand days that are weighted with weighting
418
factors to represent a whole year, it is necessary to introduce cycling conditions.
419
init These state that the storage’s initial storage level at each day Sdev,i,d has to
420
init be equal to its final level at the end of the day. In our model, Sdev,i,d is also
421
considered as a decision variable that is optimized.
TE
D
417
init Sdev,i,d,−1 = Sdev,i,d,tend = Sdev,i,d
∀dev ∈ {BAT, T ES} i, d
(53)
Charging and discharging powers are restricted with big-M formulations,
423
preventing the charging and discharging of tanks that are not installed. Appro-
424
priate values for Mdev,i
EP
422
ch/dch
426
427
428
limits given in the data sheets, whereas multiples of Q˙ DHL are used for TES
AC C
425
for BATs are the maximum charging and discharging
units.
ch chdev,i,d,t ≤ xdev,i · Mdev,i
∀dev ∈ {BAT, T ES} , i
(54)
dch dchdev,i,d,t ≤ xdev,i · Mdev,i
∀dev ∈ {BAT, T ES} , i
(55)
For TES units, the following balances are applied to determine the charging and discharging power: 20
ACCEPTED MANUSCRIPT
chTES,i,d,t =
dev ∗
i
X
XX
Q˙ dev∗ ,j,d,t
∀d, t
j
dchTES,i,d,t = Q˙ DHW + Q˙ SH d,t d,t
∀d, t
i
(57)
In these formulations, dev ∗ describes the subsets of all heat generating de-
430
vices, Q˙ DHW and Q˙ SH d,t d,t describe the building’s domestic hot water and space
431
heating demands at time t on day d. Both equations, in combination with the
432
big-M constraints, ensure that all generated and consumed heat are interchanged
433
with exactly one thermal storage tank.
M AN U
SC
429
434
Since we assume DHW to be heated up from 10◦ C to 60◦ C and both STC
435
and HP to only provide heat at 35◦ C, the remaining temperature lift has to be
436
provided by other heat generators than HP or STC. Due to the usage of typical
437
demand days, these constraints are formulated for the entire day and not for
X
each hour, allowing for a flexible charging and discharging strategy: X X X 60 − 35 X ˙ Q˙ dev,i,d,t ≥ · QDHW,d,t · xdev,i ∀dev ∈ {HP, ST C} , d 60 − 10 t i i
D
438
(58)
TE
dev∈{BOI,CF W H,CHP,EH}
Additionally, CFWHs are assumed to have a small storage volume in com-
440
parison with the TES unit, therefore the following equation limits the heat
441
output of the CFWH:
EP
439
X
Q˙ CF W H,i,d,t ≤ Q˙ DHW,d,t
∀i, d, t
(59)
i
442
443
444
445
AC C
t
(56)
RI PT
X
For BAT systems, the charging and discharging powers follow from the build-
ing’s electricity balances. We formulate one balance for the building’s plug loads and a second balance for the electricity potentially billed with a special heat pump tariff. The building’s electricity balance is written as: X X P L,imp PL PL PL PL PL PL Pd,t +PCFWH,d,t +PEH,d,t + chBAT,i,d,t = Pd,t +PPV,d,t +PBAT,d,t + PCHP,i,d,t i
i
(60)
21
∀d, t
ACCEPTED MANUSCRIPT
PL PL PL Here, Pd,t describes the house’s plug loads, PCFWH,d,t and PEH,d,t stand for
447
the amount of electricity consumed by an CFWH or EH that is not billed under
448
a special HP tariff.
RI PT
446
The second electricity balance is:
449
X
X HP,imp HP HP HP HP HP PHP,i,d,t +PCFWH,d,t +PEH,d,t = Pd,t +PPV,d,t +PBAT,d,t + PCHP,i,d,t
i
i
(61)
The CFWH’s and EH’s electricity amounts are further defined: In total,
451
the amount of electricity purchased at a potential HP tariff and the amount
452
purchased at the standard tariff, have to be equal to the electricity consumption caused by this device: X PL HP Pdev,i,d,t = Pdev,d,t + Pdev,d,t
M AN U
453
i 454
455
SC
450
∀dev ∈ {CF W H, EH} , d, t
(62)
If no HP has been purchased, the CFWH and EH cannot be active in the second electricity balance: nom X nom HP HP · xHP,i PCFWH,d,t + PEH,d,t ≤ max PCFWH,j + max PEH,j j
j
∀d, t
i
D
(63) Finally, we prevent the HP and STC from being activated if the thermal
TE
storage’s average temperature is above the set flow temperature of 35◦ C. We assume that both can provide a temperature spread of 10 K and that the storage
456
457
458
AC C
EP
is ideally stratified. X ∆T HP STES,i,d,t ≤ 1 − yHP,j,d,t · 1 − ∆T max i X ∆T STC STES,i,d,t ≤ 1 − ySTC,j,d,t · 1 − ∆T max i
∀j, d, t
(64)
∀j, d, t
(65)
2.4. Ecological constraints In line with the Paris Agreement [57], we use CO2 emissions as an index for
measuring the ecologic impact of the optimal energy systems. Annual CO2 emis-
459
sions comprise emissions from gas usage emigas , imported electricity emiel,imp
460
and negative emissions from electricity exports emiel,exp . emiann = emigas + emiel,imp − emiel,exp 22
(66)
∀d, t
ACCEPTED MANUSCRIPT
are assumed to be time-independent, the emissions from natural gas result in: emigas =
X
emigas,spec · j
X
j 463
l
X
emiel,spec · j
X
j
PL HP Elj,l + Elj,l
l
CO2 [58]: = 0.527 kgkWh tricity mix which causes emiel,spec EM
"
el,exp
emi
=
emiel,spec EM
·
X
wd · ∆t ·
d
466
(67)
(68)
Electricity fed into the public grid is expected to replace the average elec-
M AN U
465
Similarly, CO2 emissions from electricity imports are modeled: emiel,imp =
464
CHP GBOI j,l + Gj,l
RI PT
462
Considering the specific CO2 emissions emigas,spec for each tariff j, which j
SC
461
2.5. Solution algorithm
!
X
X
t
i
sell PCHP,i,d,t
+
# sell PPV,d,t
+
sell PBAT,d,t
(69)
The described model defines a mixed integer linear program that can be
468
solved with existing solvers. In this work, all computations are carried out with
469
Gurobi 7.02 and all models are set up with the corresponding Gurobi-Python
470
framework (gurobipy). We used the standard Branch-and-Cut [59] algorithms
471
from this solver.
TE
D
467
Since the calculating times of the described MILP problem easily become
473
intractable with increasing typical demand days, time resolution, and available
474
devices, we implemented a decomposition approach that has also been proposed
475
by Wakui and Yokoyama [60]. In this approach, the solution space is reduced by
477
478
479
480
AC C
476
EP
472
prescribing the device selection. Additionally, the objective function is forced to be lower than the previously determined minimum objective value. This limit on the objective additionally accelerates the solution process, since device candidates that cannot improve the objective are quickly marked as infeasible and therefore solving until optimality is omitted. 2 http://www.gurobi.com/index
23
ACCEPTED MANUSCRIPT
In our implementation of this approach, each major conventional heat gen-
482
erator (gas boiler, CHP unit and HP unit) is consecutively fixed, starting with
483
the smallest gas boiler and finishing with the largest HP unit.
484
3. Setup
RI PT
481
This section summarizes the used inputs and describes the conducted calcu-
486
lations. Further information on these inputs as well as the implemented model
487
can be found in this project’s open-source repository3 .
488
3.1. Inputs
M AN U
SC
485
489
The model’s input time series consist of solar irradiation onto the roof area
490
and the building’s energy load profiles, such as electricity demands of plug loads,
491
domestic hot water usage, and space heating loads. Additionally, the considered
492
energy conversion units are described in this subsection.
493
3.1.1. Energy load profiles
We apply the optimization model described in the previous section to three
495
newly constructed residential buildings significantly varying in size that are
496
located in different regions of Germany. The first building is a small single
497
family house located in Bavaria, the second building is a medium-sized multi
498
family house in Hamburg, and the third building represents a large apartment
499
building in Berlin. The key parameters are summarized in Table C.5. For all
500
roofs solar modules are assumed to be oriented southwards and elevated at a
501
pitch angle of 35◦ .
503
504
TE
EP
AC C
502
D
494
Time series for ambient temperature and solar irradiation for calculating
the building’s heat demand and PV as well as STC efficiencies and outputs are taken from German Test Reference Years [61] for the corresponding regions. 3 https://github.com/RWTH-EBC/BESopt
24
ACCEPTED MANUSCRIPT
The hourly space heating profiles are computed with the freely available
506
building simulation library AixLib [62]4 and the building models are param-
507
eterized with the also open-source available software package TEASER [63]5 .
508
Electricity demands for non-heating devices, appliances and lighting are com-
509
puted with a high-resolution, stochastic tool based on Richardson et al. [64].
510
Domestic hot water demand profiles are calculated with a combination of the
511
users’ occupancy based on Richardson et al. [64] and daily tap water usage
512
statistics of residential buildings developed in IEA Annex 42 [65]. The cumu-
513
lated, annual domestic hot water, space heating and electricity demands are also
514
listed in Table C.5.
M AN U
SC
RI PT
505
As mentioned previously, using full year inputs leads to computationally
516
intractable simulations. Therefore, the input time series are reduced to 5 typical
517
demand days with hourly time resolution, by using k-medoids clustering [43, 44,
518
45]. We have chosen 5 typical demand days since the original building energy
519
system optimization model also used 5 days, however these demand days were
520
determined differently [41, 42].
521
3.1.2. Energy conversion units
D
515
All energy conversion units that have been modeled are based on manu-
523
facturers’ data sheets. Price recommendations are taken from multiple online
524
retailers or original manufacturers for all devices but CHP units. For CHP units,
525
the regression curves provided by ASUE [49] have been used for estimating CHP
526
units’ investment costs.
528
529
530
531
EP
Table C.6 displays the characteristics of all used gas boilers and CHP units.
AC C
527
TE
522
In this table, all used node points for the interpolation of heat and power output as well as gas consumption during part load operation are listed. We assume that each device can operate flexibly between these given points. The index ‘mds’ refers to manufacturer’s data sheets. 4 https://github.com/RWTH-EBC/AixLib 5 https://github.com/RWTH-EBC/TEASER
25
ACCEPTED MANUSCRIPT
Table C.7 shows the considered CFWHs, EHs and HPs. The nominal heat
533
output Q˙ nom and electricity input P nom of HPs describe the device’s operation
534
at approx. 7 ◦ C outdoor temperature and 35 ◦ C flow temperature.
RI PT
532
The characteristics displayed in Table C.8 have been used for all HPs. The
536
left column represents the ambient temperature and the right columns the device
537
operation, starting with the full load operation. In each cell, the left entry
538
represents the scaled heat output and the right entry the also scaled electricity
539
consumption. The scaling factors are equal to the nominal heat outputs shown
540
in Table C.7. We assume that HPs are primarily used for covering space heating
541
and therefore provide heat at 35◦ C at all times.
543
M AN U
542
SC
535
The considered STCs are listed in Table C.9, PV modules and INVs are shown in Table C.10.
Table C.11 displays the available BAT storages and TES units. BAT’s ca-
545
pacity is given in kWh and for TES, m3 is used. Furthermore, BAT’s capacity
546
describes the effective capacity, including the maximum allowed depth of dis-
547
charge. TES units’ self-discharge is considered through the loss coefficient ϕ
548
that describes the energy loss during 24 hours of standby operation. In con-
549
trast, self-discharge is neglected for BATs [66].
550
3.1.3. Tariffs and emissions
TE
D
544
The electricity and gas tariffs’ characteristics are listed in Tables C.12. All
552
tariffs are derived from a local utility provider [67]. The specific emissions for
553
standard and HP tariffs are based on [58]. Price information and the tiered
554
pricing structure are taken from [67]. Regarding green tariffs, we assumed a
556
557
558
AC C
555
EP
551
share of 45% of renewable electricity generation and the remaining share to be generated conventionally but covered with internationally available renewable certificates [68, 69]. These remaining 55% are therefore assumed to cause local emissions that are equal to the average electricity mix. For eco gas, a 10% share
559
of biogas is assumed to increase the variable costs by 0.0060 Euro/kWh [70]. As
560
shown in both tables, standard tariffs are slightly less expensive than eco-tariffs,
561
but cause significantly more CO2 emissions. The implemented HP tariff further
26
ACCEPTED MANUSCRIPT
offers a strong economic incentive compared to the other tariffs.
563
3.2. Economic parameters
RI PT
562
Table C.13 lists the economic parameters used in this study. The price
565
change factors for electricity and gas are derived by linear regressions from the
566
average German prices between 2008 and 2016 [71]. Similarly, the price change
567
factor for EEX compensation is derived by linear regression of the CHP indexes
568
of the past ten years [53]. The expected lifetimes of the devices are based on
569
VDI 2067 [46].
570
3.3. Calculations
M AN U
SC
564
571
The calculations conducted in this work aim at comparing the influence of
572
the discrete modeling of storage units, assessing the importance of modeling
573
subsidies and market characteristics and analyzing the robustness of our model.
574
3.3.1. Discrete storage modeling
In order to evaluate the importance of a discrete storage selection, we con-
576
duct a total of six optimizations for each building. These consist of a cost
577
minimization for the new and original model as well as a recalculation. Since
578
the storage units are expected to deviate due to the different modeling, the re-
579
calculation of the original model is conducted, enforcing the optimal solution of
580
the new model. Additionally, a second set of three optimizations is conducted
581
with an enforced reduction of CO2 emissions by 20%.
582
3.3.2. Importance of subsidy modeling
584
585
586
TE
EP
AC C
583
D
575
For assessing the importance of accounting for subsidies, we compare the
results of our model for the cost minimization and enforced CO2 reduction with
a model neglecting all previously mentioned subsidies and regulations. In this case, ceeg and esub are set to zero, and the feed-in limit of 70% of the installed
587
peak power for PV modules is omitted. Additionally, we assume that electricity
588
from PV units is sold at the same market price as electricity from CHP units
589
and that only standard tariffs are available. 27
ACCEPTED MANUSCRIPT
590
Thus, we analyze and compare three different models. The total amount of constraints and variables for these models are summarized in Table C.14.
592
3.3.3. Sensitivity analysis
RI PT
591
The model relies on a number of uncertain factors, for instance weather
594
conditions, user behavior, energy tariffs, device operation and investment costs.
595
Weather conditions and user behavior both strongly affect the energy load
596
profiles, such as household electricity, domestic hot water and space heating
597
demands. These effects can be analyzed and taken into consideration through
598
various different methods, such as weather data from different years, extreme
599
weather scenarios, and complex user behavior models. In this work, we account
600
for these effects, in a simplified manner by linearly scaling the original demands
601
with 0.95 (-5% demands scenario) and 1.05 (+5% demands scenario).
M AN U
SC
593
Additionally, the chosen tariffs are expected to significantly impact the opti-
603
mization results. Therefore, we also analyze a reduced and a high tariff scenario,
604
by linearly scaling the original electricity and gas tariffs’ variable costs with 0.95
605
(-5% tariffs scenario) and 1.05 (+5% tariffs scenario).
D
602
Device operation is uncertain since it depends on the quality of the instal-
607
lation, for instance the system’s hydraulic balance. Investment and installation
608
costs are also considered to be uncertain, since they strongly vary regionally and
609
they are typically highly influenced by the customer’s market power and nego-
610
tiation abilities. Since we assume that device operation is of less importance
611
than the other influences and that investment costs can be determined precisely
612
when applying the model for real life applications based on price inquiries, we
614
615
616
617
EP
AC C
613
TE
606
do not include these factors in our sensitivity analysis. Similar to the influence analysis of the discrete storage model, we also con-
duct a new optimization run for each scenario of the uncertainty analysis as well as a recalculation that enforces the base case’s results given the changed demands or tariffs.
28
ACCEPTED MANUSCRIPT
618
3.4. Computing hardware R R For all optimizations, we used a Windows 7 computer with an Intel Xeon
620
E5-2630 v2 CPU utilizing 6 threads and 32 GB of RAM. All optimizations are
621
solved to an optimality gap of 1% and all recalculations are solved with a gap
622
of 0.1%. Since most inputs are uncertain and the model also still presents
623
simplifications that will be addressed in Section 4.5, an optimality gap of 1% is
624
assumed to be reasonable.
625
4. Results
SC
RI PT
619
First, the discrete storage tank modeling and the importance of accounting
627
for subsidies are evaluated for all three buildings. Subsequently, the sensitivity
628
analysis is presented and the limitations of our model are discussed.
629
4.1. Single family house
M AN U
626
The key results of cost minimizations and forced CO2 reductions for the
631
single family house are summarized in Table C.15. The left part shows the cost
632
minimization results whereas the right part lists the key findings of the forced
633
CO2 reductions. The columns labeled ‘new model’ stand for our developed
634
model, whereas ‘original’ marks the original model based on Wakui et al. [42] and
635
’no subsidies’ represents our modeling approach without considering subsidies
636
and regulations. The original formulation calculations are further split into a
637
real optimization (opt.) of the energy system design and its operation as well
638
as a recalculation (rec.). In the recalculation, the optimal energy system of
639
the new model is imposed on this model, leaving the system’s operation as the
641
642
643
644
TE
EP
AC C
640
D
630
remaining decision variables. In all cases, neither BATs, CFWHs, CHPs nor HPs have been chosen. Ad-
ditionally, the installed PV capacities are below 10 kW, therefore, no EEG levy has to be paid. Furthermore, all calculations lead to standard gas and electricity tariffs.
645
The influence of discrete storage modeling can be seen by comparing the
646
new model with the original formulation. For the single family house, the cost 29
ACCEPTED MANUSCRIPT
optimal energy system consists of a small boiler that is supported by one STC
648
module of type 3. The remaining roof area is covered with 37.62 m2 of PV
649
units, which is equivalent to 23 modules of type 3. In the new model with
650
discrete storage tank modeling, a tank with 0.49 m3 volume is selected, whereas
651
the continuous storage modeling in the original model leads to a smaller TES
652
with 0.33 m3 . Consequently, the original model requires lower investment costs
653
but leads to higher demand related costs since less heat from the STC can be
654
utilized. The recalculation shows that the modeling of TES units’ investment
655
costs causes a deviation of 1.88 Euro/a. In total, the costs of the new and original
656
model differ by less than 0.5%. Due to the large PV area and the resulting high
657
amount of PV feed-in, negative CO2 emissions occur that consequently lead to
658
the same energy systems when enforcing a relative reduction of CO2 emissions
659
by 20%.
M AN U
SC
RI PT
647
In contrast, without subsidies an additional EH is installed and the PV
661
area is strongly reduced. As a consequence, less PV is exported leading to lower
662
negative CO2 emissions. When enforcing the CO2 reduction, TES size as well as
663
the amount of PV modules and STCs are increased. The larger TES is necessary
664
for integrating more heat from the STCs. Since more heat is generated through
665
solar energy, less gas is required from the boiler which switches the gas tariff from
666
level 2 to level 1, strongly reducing the fixed metering costs. Table C.15 also
667
shows that without subsidies, the revenues from electricity feed-in are strongly
668
reduced, since less PV modules are installed and feed-in is only remunerated with
669
the average market rate and not the high remuneration according to the EEG.
670
The comparison with the original model shows that accounting for available
672
673
674
675
TE
EP
AC C
671
D
660
subsidies and market characteristics leads to cost benefits of 278.26 Euro/a (14.8%) and 285.46 Euro/a (15.2%) for the forced CO2 reductions. For the single family house, the new model requires approx. 660 seconds of
calculating time, the original model 1340 seconds and the new model without subsidies needs 750 and 850 seconds.
30
ACCEPTED MANUSCRIPT
676
4.2. Multi family house Similar to the single family house, Table C.16 displays the results for the
678
multi family building. The original and new model formulations again lead to
679
very similar results and only slightly differ in the TES sizes for both, the cost
680
minimization and the CO2 reduction. The optimal energy systems consist of a
681
medium sized CHP unit with 12.5 kW heat output in combination with an EH
682
that functions as a backup unit and is necessary for meeting the design heat load.
683
Additionally, a TES unit with 0.89-0.98 m3 and a large PV area are installed. In
684
the cost minimization case, the PV area has a peak output of 9.99 kW, therefore
685
no EEG levy has to be paid. In contrast, when enforcing the CO2 reduction,
686
the PV area is enlarged, leading to EEG levies of approx. 460 Euro/a (roughly
687
11% of the total annual costs). The comparison of the storage modules shows
688
that the new model predicts slightly larger storage capacities. The modeling of
689
the TES investment costs leads to deviations of less than 3.77 Euro/a and total
690
costs also only deviate by less than 15.45 Euro/a (0.21%).
M AN U
SC
RI PT
677
The no subsidies cases lead to energy systems that strongly utilize gas boil-
692
ers for this building. The cost minimal solution uses a 20.3 kW gas boiler in
693
combination with an 8 kW EH and a 0.75 m3 TES unit. Additionally, a 22 kWh
694
BAT is installed and the entire available roof area is covered with PV modules.
695
When enforcing CO2 reductions, a small CHP unit is installed and 4 modules of
696
STC type 3 replace a portion of the PV area. The cost optimal solution strongly
697
focuses on self-consumption, therefore the BAT and EH are installed to capital-
698
ize on PV generated electricity. By using electricity for heating purposes, heat
699
can be produced at 0.0380 Euro/kWh (missed remuneration for avoided grid
701
702
703
TE
EP
AC C
700
D
691
feed-in), instead of more than 0.0615 Euro/kWh when using the gas boiler. Unlike for the single family house, the consideration of subsidies increases
the carbon footprint in the cost minimization. More PV modules are built and less grid electricity is needed when neglecting subsidies in this case, leading
704
to lower total emissions. This is obviously in contradiction with the intention
705
of the modeled subsidies. However, when enforcing CO2 reductions, more PV
706
units are used in the new and original model, which in turn leads to lower 31
ACCEPTED MANUSCRIPT
CO2 emissions than in the no subsidies calculation. Additionally, the solutions
708
of the CO2 reduced scenario are cheaper and generate less CO2 than the cost
709
minimum without these subsidies. This illustrates that subsidies still support
710
CO2 reductions and provide benefits for the energy system’s owner.
RI PT
707
The calculating times for the multi family house are approx. 2935 and 1580
712
seconds with the new model, 1970 and 710 seconds with the original formulation
713
and 1450 as well as 1780 seconds without considering subsidies and market
714
characteristics.
715
4.3. Apartment building
M AN U
SC
711
Table C.17 displays the results for the apartment building. The cost optimal
717
energy systems obtained when considering subsidies and market characteristics,
718
comprise a medium sized CHP unit that is supported by a boiler and an EH
719
to meet the design heat load. The TES sizes vary between 1.50 m3 in the new
720
model and 1.36 m3 in the original model formulation. In all cases, the available
721
roof area is covered with PV modules exclusively, exceeding the 10 kW threshold
722
which leads to EEG levy charges between approx. 750 and 850 Euro/a. When
723
requiring CO2 reductions, a HP unit replaces the EH. Additionally, the gas
724
boiler is slightly enlarged for covering the design heat load. In contrast to the
725
other simulations, the electricity tariff is also used as a measure for reducing
726
CO2 emissions. When reducing CO2 emissions, an eco tariff is used, whereas
727
in the cost optimal solutions a standard tariff is employed. In the reduced CO2
728
case, the sizes of the CHP units and consequently also of the TES units are
729
enlarged. The original model formulation requires a TES with 1.85 m3 volume,
731
732
733
TE
EP
AC C
730
D
716
whereas the new formulation uses a 2.00 m3 tank. Additionally, 1 STC module
is installed in the new model and 2 modules are used in the optimal energy system based on the original formulation. The different tank sizes lead to errors in the estimation of investment costs of approx. 15 Euro/a, which is considered
734
negligible in comparison with the total investment costs of approx. 6300 Euro/a.
735
Furthermore, the inaccuracy resulting from scaling a representative TES unit is
736
also not significant since the models differ by at most 0.1%. 32
ACCEPTED MANUSCRIPT
Without considering subsidies, a smaller CHP unit is required. For compen-
738
sation, a larger boiler is installed. As a consequence, the TES volume can be
739
reduced to only 0.98 m3 . Like for the multi family building, the case without
740
subsidies leads to the installation of a BAT. For this building, approx. 128 m2
741
of the roof area are covered with PV units and additionally, one STC module
742
is installed. When forcing CO2 reductions, the PV area is maxed out and more
743
PV is fed into the grid. The installation of additional 39 m2 of PV modules
744
(24 modules) only increases the total costs by 60 Euro/a, indicating that PV
745
modules are priced competitively under the used inputs and assumptions. In
746
both calculations, the overall CO2 emissions are significantly higher than in the
747
cases when accounting for subsidies. Therefore, the governmental subsidies and
748
market characteristics lead to the intended goal of significantly reducing CO2
749
emissions for this building.
M AN U
SC
RI PT
737
For the apartment building, the new model requires approx. 3285 and 4615
751
seconds, the original model 3135 and 4840 seconds, and the new model without
752
subsidies 530 and 1275 seconds. Overall, the discrete modeling of storage devices
753
does not appear to strongly influence the calculating times. When comparing
754
the calculating times for all presented optimization runs by using the geometric
755
mean, the new model improves the run times by 5%6 . However, considering
756
subsidies and market characteristics strongly increases the computing times.
757
In our analysis, the run times are increased by 71% when using the geometric
758
mean.
EP
TE
D
750
These results are counterintuitive considering the number of constraints and
760
variables in each model. All three models have roughly the same amount of
761
762
763
AC C
759
binary variables. The original model has the lowest number of constraints and total variables, whereas the new formulation requires the highest amount of both. In combination, this often suggests that the original model would have 6 The
geometric mean is used since the calculating times are on different ranges. When
comparing both with a different metric, average relative deviations, the original model has 11% lower run times.
33
ACCEPTED MANUSCRIPT
the fastest solving times. However, we believe that the solution times can be ex-
765
plained best by considering the structure of the constraint matrix. Accounting
766
for a discrete storage model hardly affects the structure of the constraint ma-
767
trix, therefore the calculating times are similar in the new and original model.
768
However, the introduction of subsidies and market characteristics adds dense
769
constraints to the model, such as Equations 20, 21, 27 and 31. According to
770
the Gurobi website [72], Gurobi and other commercial MILP solvers are able to
771
strongly exploit sparse constraint matrices, leading to low calculating times.
772
4.4. Sensitivity analysis
M AN U
SC
RI PT
764
The results of the sensitivity analysis are shown in Table C.18. For the single
774
family house, only a demand reduction leads to an adjusted cost-optimal energy
775
system. In this case, the storage unit can be sized slightly smaller. Furthermore,
776
in this case it is necessary to allow for installing a small EH, because due to the
777
reduced demands and hourly time discretization, the minimum gas boiler heat
778
output would already provide too much heat that would not allow for fulfilling
779
the storage’s cycling conditions (Equation 53). The resulting cost difference for
780
the -5% demands scenario is approx. 8.30 Euro/a which is equivalent to 0.5%.
781
In the other cases related to the SFH, the base case’s optimal energy system is
782
always a robust solution that is not altered.
TE
D
773
The resulting cost optimal energy systems for the multi family house differ
784
for the -5% demands and -5% tariffs scenarios. With lower demands, the CHP
785
unit becomes less attractive due to reduced operating times and is therefore re-
786
placed by a gas boiler. Additionally, the TES volume is decreased and two STC
788
789
790
AC C
787
EP
783
modules with a total of 4.62 m2 as well as a larger PV area are installed. In this manner, the no longer available electricity generation from CHP is partly compensated with PV modules, which reduces electricity purchases from the grid. This case poses the largest deviation between the resulting total annual costs of
791
the optimization and recalculation, amounting to 103.48 Euro/a, approx. 1.5%.
792
A second deviation occurs for the -5% tariffs case. In this setting, a similar
793
optimal energy system is found as in the scenario with reduced demands. The 34
ACCEPTED MANUSCRIPT
initially installed CHP unit becomes less economic since savings from generating
795
electricity in comparison to imports from the grid are reduced. The resulting
796
cost deviation is 0.7%.
RI PT
794
The sensitivity analysis reveals that a 5% increase in demands makes a
798
larger CHP unit preferable for the apartment building. In this case, the TES
799
unit is increased to 2.00 m3 in order to provide more operating flexibility and
800
increase the operating time of the prime mover. The costs of optimization and
801
recalculation differ by 0.5% in this setting.
SC
797
Overall, the results of the base case appear to be relatively robust. The
803
optimal energy systems are only changed in 4 out of the investigated 12 cases.
804
The prime mover is increased in capacity once and completely replaced in two of
805
these cases. The maximum cost deviation between recalculation and optimiza-
806
tion is 1.5% in the multi family case with 5% demand reduction. On average,
807
the costs between optimization and recalculation differ by 0.3%.
M AN U
802
Interestingly, despite the fact that the optimization run is always conducted
809
with a tolerated optimality gap of 1% and the recalculation has a gap of 0.1%,
810
the solutions only marginally differ by at most 0.04% if the same energy systems
811
are chosen. This suggests that the solver converges to an (almost) optimal
812
solution even at a higher tolerated gap.
813
4.5. Limitations and future work
816
817
818
819
820
821
TE
EP
815
This section critically assesses the limitations of the developed model and presents possible improvements. This paper focuses on the optimal design, sizing and operation of building
AC C
814
D
808
energy systems for residential buildings. The developed model has been applied to newly constructed buildings. The model could be applied to existing buildings with already installed devices, however this would require multiple runs in which either the heat distribution system (radiators, floor heating) would be used after refurbishment or be replaced, too.
822
Since the model is currently only intended for single buildings, regional ef-
823
fects are neglected. Considering the obtained results and assuming a wide ap35
ACCEPTED MANUSCRIPT
plication of our findings, large areas of PV units would provide high amounts
825
of electricity during sunny periods and feed them into the grid. This could
826
potentially be harmful for existing distribution grids, leading to congestion and
827
voltage fluctuations in case of abruptly changing weather conditions, e.g. caused
828
by clouds. Considering multiple buildings simultaneously could potentially lead
829
to smarter subsystems consisting of local generation and consumption units as
830
well as electricity storages.
RI PT
824
Currently, the model has been used with hourly inputs. Due to the already
832
relatively long computing times and the low availability of weather data with
833
higher resolution, hourly inputs have been used as a trade-off solution. However,
834
using hourly averaged time series strongly benefits solar generation units, since
835
fluctuations in the solar irradiation as well as fluctuating local demands are
836
balanced, leading to higher rates of self-consumption. Since the share of self-
837
consumed electricity tends to be overestimated, grid imports and consequently
838
overall costs are expected to be underestimated.
M AN U
SC
831
The presented model does not account for installation and assembly costs in
840
detail. These costs could be accounted for in a higher level of detail, e.g. module
841
specific installation costs as well as module unspecific costs, such as call out
842
charges. Including these costs would likely affect the results, in particular the
843
installation of one or two STCs would become less attractive. These costs can
844
hardly be estimated from an academic point of view since they strongly depend
845
on the building’s location, installation service, etc. However, such costs could
846
be specified precisely by installation companies using tools like our developed
847
model.
849
850
851
TE
EP
AC C
848
D
839
In addition, there are large uncertainties regarding the modeling of specific
tariff parameters and CO2 emissions. The pricing structure has largely been based on our local utility service provider, therefore tariffs could strongly deviate in other regions. Furthermore, we modeled the eco tariff for electricity assuming
852
that 45% of the total electricity is generated with RES. When choosing this
853
share of renewables, we assumed that the remaining part would be generated
854
conventionally and covered with renewable certificates [68, 69]. This assumption 36
ACCEPTED MANUSCRIPT
might not be valid for all tariffs. Therefore, future studies could investigate the
856
influence of tariffs that exclusively offer locally generated electricity from RES.
857
Moreover, we used constant emission factors for gas and electricity in this study,
858
since such data is typically available and represents current balancing standards.
859
However, accurately assessing the emissions caused by the energy system and
860
its operation, requires dynamic emission factors that reflect the entire electricity
861
generation within the market.
RI PT
855
An obvious weakness of our model is the assumption of perfect knowledge of
863
upcoming demands. We assume electricity and thermal demands as well as solar
864
irradiation to be known precisely for each considered time step. Additionally, we
865
assume these inputs to be constant for the entire observation period. In reality,
866
these demands significantly change from year to year, since the building’s usage
867
and ambient conditions change. Furthermore, even short-term load forecasts
868
suffer from inaccuracies, leading to suboptimal device scheduling. This problem
869
could be overcome by using a multi-level optimization approach [73]. On a top
870
level, the device selection could be handled, whereas a rolling horizon scheme
871
with either robust or stochastic optimization, can be used on a lower level to
872
optimize the operation.
TE
D
M AN U
SC
862
Related to the previous limitation, a longer observation period would present
874
an interesting extension for this model. In this manner, modified building usage,
875
e.g. due to changing family structures, can be taken into account and roadmaps
876
for the installation and replacement of energy system components could be
877
developed.
879
880
881
882
Furthermore, the model could be extended in future works by accounting
AC C
878
EP
873
for start, stop and ramping constraints. Currently, there are neither minimum activation nor minimum turn-off times implemented. Future works could integrate such minimum time durations in which the devices has to stay activated or deactivated in order to reduce wear and tear. Additionally, such device wear
883
and tear is currently not considered. Since the observation period already covers
884
multiple years, accounting for reduced efficiencies due to these effects is likely
885
to affect the results. 37
ACCEPTED MANUSCRIPT
The model is intended for optimizing the building’s energy system. How-
887
ever, accounting for investments into passive components such as the building’s
888
envelope is expected to also affect the energy system. Future works could there-
889
fore integrate design decisions regarding envelope components into the current
890
model.
891
5. Conclusions
SC
RI PT
886
In this paper, a MILP for the optimal structural design, sizing and operation
893
of building energy systems has been developed. The model enhances existing
894
formulations by considering specific German regulations and market character-
895
istics, and by accounting for multiple gas and electricity tariffs. Additionally,
896
we extend previous approaches by a discrete sizing of storage units.
M AN U
892
The model has been applied to three newly constructed residential buildings
898
in different locations in Germany. Accounting for a discrete sizing of storage
899
units has marginally impacted the optimal energy systems and estimated total
900
costs. However, due to the device-specific modeling, the optimal energy system
901
is ensured to be available for purchase in reality, whereas the original formulation
902
can lead to intermediate sizes that cannot be purchased (e.g. a 1.36 m3 TES
903
instead of a 0.98 m3 or 1.50 m3 device which are available). Despite the fact that
904
the proposed formulation increases the amount of constraints and variables, the
905
effect on computing times is negligible. Depending on the used metric, either
906
the new model or the original formulation is beneficial regarding run times.
908
909
910
911
912
TE
EP
On the other hand, accounting for subsidies and market characteristics has
AC C
907
D
897
shown to strongly influence the optimal energy systems. When neglecting these constraints, less PV modules but more STCs are installed. Additionally, BATs are purchased for using electricity from PV locally, reducing grid dependence in contrast to feeding PV into the grid for a low remuneration. Furthermore, CHP units become less attractive when not considering subsidies.
913
The conducted sensitivity showed that the computed optimal energy systems
914
are relatively robust regarding demand and tariff changes. Although the tariffs
38
ACCEPTED MANUSCRIPT
and demands have been varied by 5%, the total costs only deviate by 1.5% when
916
comparing the original and the new optimal solution. Both optimal solutions
917
differ in 4 out of the 12 investigated cases and in only two of these cases, the
918
energy system is changed substantially.
RI PT
915
Several limitations of our model have been identified and discussed. The
920
most important weaknesses are the focus on individual buildings, the assump-
921
tion of perfect predictions as well as the low level of detail when modeling instal-
922
lation costs. In this regard, future works could extend the model to small city
923
districts, include a multi-level optimization for dealing with uncertain demand
924
predictions and accounting for different types of installation costs. Furthermore,
925
the model can be improved by considering longer observation periods, modeling
926
start-up, shut-down and ramping constraints as well as devices’ wear and tear.
927
Additionally, a more holistic approach can be implemented in future works that
928
accounts for also optimizing passive components such as the building’s enve-
929
lope.
930
Acknowledgments
M AN U
D
TE
Initiative “Energy System 2050 - A Contribution of the Research Field Energy”.
EP
932
This work was supported by the Helmholtz Association under the Joint
AC C
931
SC
919
39
933
Appendix A. Energy balances
934
Appendix B. Additional equations
RI PT
ACCEPTED MANUSCRIPT
Modeling of investment costs: cinv dev = CRF ·
X xdev,i · (1 − rvdev,i ) · cinv dev,i
∀ dev ∈ devs\ {P V, ST C}
i
cinv dev
SC
(B.1)
X = CRF · zdev,i · (1 − rvdev,i ) · cinv dev,i
∀ dev ∈ {P V, ST C}
cinv =
X
M AN U
i
cinv dev
dev 935
(B.2) (B.3)
At most one type of each device may be chosen: X
xdev,i ≤ 1
i
∀dev
(B.4)
The available roof area cannot be exceeded:
dev
∀dev ∈ {P V, ST C} , i
zdev,i · Adev,i ≤ Amax
i
(B.5) (B.6)
TE
XX
Amax Adev,i
D
zdev,i ≤ xdev,i ·
EP
Modeling of operation and maintenance costs: infl co&m · CRF · dev = b
X
o&m xdev,i · cinv dev,i · fdev,i
∀ dev ∈ devs\ {CHP, P V, ST C}
AC C
i
infl co&m · CRF · dev = b
infl co&m · CRF · CHP = b
936
X
(B.7) o&m
zdev,i · cinv dev,i · fdev,i
∀ dev ∈ {P V, ST C}
i
(B.8) XXX i
d
o&m PCHP,i,d,t · ∆t · fCHP,i
(B.9)
t
Heating devices can only be switched on, if they have been purchased: ydev,i,d,t ≤ xdev,i
∀dev ∈ {BOI, CF W H, CHP, EH, HP } , i, d, t 40
(B.10)
ACCEPTED MANUSCRIPT
Piecewise linearization of the performance charts of boilers, CHP units and
Q˙ dev,i,d,t =
X
wdev,i,d,t,k · Q˙ mds dev,i,d,t,k
RI PT
HP units: ∀dev ∈ {BOI, CHP, HP } , i, d, t
k
(B.11)
E˙ dev,i,d,t =
X
mds wdev,i,d,t,k · E˙ dev,i,d,t,k
∀dev ∈ {BOI, CHP, HP } , i, d, t
Pdev,i,d,t =
X
mds wdev,i,d,t,k · Pdev,i,d,t,k
∀dev ∈ {BOI, CHP, HP } , i, d, t
wdev,i,d,t,k
k
Part load of CFWHs and EHs: Q˙ dev,i,d,t ≤ Q˙ nom dev,i
(B.15)
∀dev ∈ {CF W H, EH} , i, d, t
(B.16)
Q˙ STC,i,d,t ≤ ηSTC,i,d,t · zSTC,i · ASTC,i · Id,t
940
941
942
(B.17)
∀i, d, t
(B.18)
AC C
PPV,i,d,t ≤ ηPV,i,d,t · zPV,i · APV,i · Id,t · η¯INV
939
∀i, d, t
Power generation of PV:
EP
938
(B.14)
∀dev ∈ {CF W H, EH} , i, d, t
TE
Heat output of STC:
(B.13)
D
Q˙ dev,i,d,t = Pdev,i,d,t 937
M AN U
ydev,i,d,t =
(B.12)
∀dev ∈ {BOI, CHP, HP } , i, d, t
k
X
SC
k
References
[1] Directive 2010/31/EU on the energy performance of buildings - 19 May 2010, Official Journal of the European Union L 153/13.
[2] Federal Ministry for Economic Affairs and Energy, Germany’s new energy
943
policy. Heading towards 2050 with secure, affordable and environmentally
944
sound energy (Apr. 2012).
41
ACCEPTED MANUSCRIPT
[3] M. M. Maghanki, B. Ghobadian, G. Najafi, R. J. Galogah, Micro combined
946
heat and power (MCHP) technologies and applications, Renewable and Sus-
947
tainable Energy Reviews 28 (2013) 510–524. doi:10.1016/j.rser.2013.07.053.
948
URL
949
S136403211300498X
RI PT
945
http://www.sciencedirect.com/science/article/pii/
[4] A. Peacock, M. Newborough, Impact of micro-CHP systems on domestic
951
sector CO2 emissions, Applied Thermal Engineering 25 (17-18) (2005)
952
2653–2676. doi:10.1016/j.applthermaleng.2005.03.015.
953
URL
954
S1359431105001018
SC
950
M AN U
http://www.sciencedirect.com/science/article/pii/
955
[5] D. Haeseldonckx, L. Peeters, L. Helsen, W. D’haeseleer, The impact of
956
thermal storage on the operational behaviour of residential chp facilities
957
and the overall co2 emissions, Renewable and Sustainable Energy Reviews
958
11 (6) (2007) 1227–1243. doi:10.1016/j.rser.2005.09.004.
959
URL
960
S1364032105001036
D
http://www.sciencedirect.com/science/article/pii/
[6] I. Staffell, D. Brett, N. Brandon, A. Hawkes, A review of domestic
962
heat pumps, Energy & Environmental Science 5 (11) (2012) 9291–9306.
963
doi:10.1039/C2EE22653G.
964
URL
965
c2ee22653g#!divAbstract
967
968
969
970
EP
http://pubs.rsc.org/en/Content/ArticleLanding/2012/EE/
[7] S. A. Kalogirou, Solar thermal collectors and applications, Progress
AC C
966
TE
961
in
Energy
and
Combustion
Science
30
(3)
(2004)
231–295.
doi:10.1016/j.pecs.2004.02.001. URL
http://www.sciencedirect.com/science/article/pii/
S0360128504000103
971
[8] B. Parida, S. Iniyan, R. Goic, A review of solar photovoltaic technolo-
972
gies, Renewable and Sustainable Energy Reviews 15 (2011) 1625–1636.
973
doi:10.1016/j.rser.2010.11.032. 42
ACCEPTED MANUSCRIPT
974
URL
975
S1364032110004016
RI PT
http://www.sciencedirect.com/science/article/pii/
976
[9] Y. Tian, C. Zhao, A review of solar collectors and thermal energy
977
storage in solar thermal applications, Applied Energy 104 (2013) 538–553.
978
doi:10.1016/j.apenergy.2012.11.051.
979
URL
980
S0306261912008549
SC
http://www.sciencedirect.com/science/article/pii/
[10] R. Luthander, J. Wid´en, D. Nilsson, J. Palm, Photovoltaic self-
982
consumption in buildings: A review, Applied Energy 142 (2015) 80–94.
983
doi:10.1016/j.apenergy.2014.12.028.
984
URL
985
S0306261914012859
986
M AN U
981
http://www.sciencedirect.com/science/article/pii/
[11] Act for the development of renewable energy sources (original German
987
title: Gesetz f¨ ur den Ausbau erneuerbarer Energien) (Jul. 2014).
988
URL
989
gesamt.pdf
D
http://www.gesetze-im-internet.de/bundesrecht/eeg_2014/
[12] Act on Combined Heat and Power Generation (original German title:
991
Gesetz f¨ ur die Erhaltung, die Modernisierung und den Ausbau der Kraft-
992
W¨ arme-Kopplung) (Dec. 2015).
993
URL https://www.gesetze-im-internet.de/bundesrecht/kwkg_2016/
994
gesamt.pdf
996
997
998
999
1000
EP
[13] Program Renewable Energies Storage (original German title:
AC C
995
TE
990
KfW-
Programm Erneuerbare Energien “Speicher”) (Mar. 2016). URL http://tinyurl.com/battery-subsidies
[14] Energy Tax Act (original German title: Energiesteuergesetz) (Dec. 2015). URL http://www.gesetze-im-internet.de/bundesrecht/energiestg/ gesamt.pdf
43
ACCEPTED MANUSCRIPT
[15] E. S. Barbieri, P. R. Spina, M. Venturini, Analysis of innovative micro-chp
1002
systems to meet household energy demands, Applied Energy 97 (2012)
1003
723–733. doi:10.1016/j.apenergy.2011.11.081.
1004
URL
1005
S0306261911007872
RI PT
1001
http://www.sciencedirect.com/science/article/pii/
[16] E. S. Barbieri, F. Melino, M. Morini, Influence of the thermal
1007
energy storage on the profitability of micro-chp systems for resi-
1008
dential building applications,
1009
doi:10.1016/j.apenergy.2012.01.001.
1010
URL
1011
S0306261912000050
Applied Energy 97 (2012) 714–722.
http://www.sciencedirect.com/science/article/pii/
[17] M. Bianchi,
M AN U
1012
SC
1006
A. D. Pascale,
F. Melino,
Performance analysis of
1013
an integrated chp system with thermal and electric energy stor-
1014
age for residential application, Applied Energy 112 (2013) 928–938.
1015
doi:10.1016/j.apenergy.2013.01.088.
1016
URL
1017
S0306261913001074
TE
D
http://www.sciencedirect.com/science/article/pii/
[18] Y. Yang, S. Zhang, Y. Xiao, Optimal design of distributed energy resource
1019
systems coupled with energy distribution networks, Energy 85 (2015)
1020
433–448. doi:10.1016/j.energy.2015.03.101.
1021
URL
1022
S0360544215004107
1024
1025
1026
http://www.sciencedirect.com/science/article/pii/
AC C
1023
EP
1018
[19] R. Loulou, U. Remne, A. Kanudia, A. Lehtila, G. Goldstein, Documentation for the TIMES Model - Part I, energy Technology Systems Analysis Programme (Apr. 2005). URL http://iea-etsap.org/docs/TIMESDoc-Intro.pdf
1027
[20] M. Stadler, DER-CAM Decision Support Tool for Decentralized En-
1028
ergy Systems, https://building-microgrid.lbl.gov/sites/default/
44
ACCEPTED MANUSCRIPT
files/projects/DER-CAMPresentation%2012%20May%202016.pdf (May
1030
2016).
1031
RI PT
1029
[21] M. B. Blarke, COMPOSE: Compare Options for Sustainable Energy.
1032
(2008).
1033
URL http://energyinteractive.net/COMPOSE.ashx
[22] M. B. Blarke, K. Yazawa, A. Shakouri, C. Carmo, Thermal battery
1035
with CO2 compression heat pump: Techno-economic optimization of a
1036
high-efficiency Smart Grid option for buildings, Energy and Buildings 50
1037
(2012) 128–138. doi:10.1016/j.enbuild.2012.03.029.
1038
URL
1039
S0378778812001776
M AN U
SC
1034
http://www.sciencedirect.com/science/article/pii/
1040
[23] E. Merkel, D. Fehrenbach, R. McKenna, W. Fichtner, Modelling decen-
1041
tralised heat supply: An application and methodological extension in
1042
TIMES, Energy 74 (2014) 592–605. doi:10.1016/j.energy.2014.06.060.
1043
URL
1044
S0360544214007592
D
http://www.sciencedirect.com/science/article/pii/
[24] K. A. Pruitt, R. J. Braun, A. M. Newman, Evaluating shortfalls in mixed-
1046
integer programming approaches for the optimal design and dispatch
1047
of distributed generation systems, Applied Energy 102 (2013) 386–398.
1048
doi:10.1016/j.apenergy.2012.07.030.
1049
URL
1050
S0306261912005521
1052
1053
1054
1055
1056
EP
http://www.sciencedirect.com/science/article/pii/
AC C
1051
TE
1045
[25] H. Ren, W. Gao, Y. Ruan, Optimal sizing for residential CHP system,
Applied
Thermal
Engineering
28
(5-6)
(2008)
514–523.
doi:10.1016/j.applthermaleng.2007.05.001. URL
http://www.sciencedirect.com/science/article/pii/
S1359431107001810 [26] K.-U. Klatt, W. Marquardt, Perspectives for process systems engineering
45
ACCEPTED MANUSCRIPT
1057
- personal views from academia and industry, Computers & Chemical En-
1058
gineering 33 (3) (2009) 536–550. doi:10.1016/j.compchemeng.2008.09.002.
1059
URL
1060
S0098135408001737
RI PT
http://www.sciencedirect.com/science/article/pii/
[27] G. Abdollahi, M. Meratizaman, Multi-objective approach in ther-
1062
moenvironomic optimization of a small-scale distributed CCHP system
1063
with risk analysis, Energy and Buildings 43 (11) (2011) 3144–3153.
1064
doi:10.1016/j.enbuild.2011.08.010.
1065
URL
1066
S0378778811003446
SC
1061
M AN U
http://www.sciencedirect.com/science/article/pii/
1067
[28] E. Barbieri, Y. Dai, M. Morini, M. Pinelli, P. Spina, P. Sun, R. Wang,
1068
Optimal sizing of a multi-source energy plant for power heat and cool-
1069
ing generation, Applied Thermal Engineering 71 (2) (2014) 736–750.
1070
doi:10.1016/j.applthermaleng.2013.11.022.
1071
URL
1072
S1359431113008119
D
http://www.sciencedirect.com/science/article/pii/
[29] B. Pickering, S. Ikeda, R. Choudhary, R. Ooka, Comparison of Metaheuris-
1074
tic and Linear Programming Models for the Purpose of Optimising Build-
1075
ing Energy Supply Operation Schedule, in: P. K. Heiselberg (Ed.), CLIMA
1076
2016 - proceedings of the 12th REHVA World Congress, Vol. 6, Aalborg
1077
University, Department of Civil Engineering, 2016, pp. 1–10.
1078
URL http://vbn.aau.dk/files/233775414/paper_529.pdf
1080
1081
1082
EP
AC C
1079
TE
1073
[30] P. Voll, Automated optimization based synthesis of distributed energy supply systems, Ph.d. thesis, RWTH Aachen University (2013). URL http://publications.rwth-aachen.de/record/228954
[31] A. Ashouri, S. S. Fux, M. J. Benz, L. Guzzella, Optimal design and
1083
operation of building services using mixed-integer linear programming
1084
techniques, Energy 59 (2013) 365–376. doi:10.1016/j.energy.2013.06.053.
46
ACCEPTED MANUSCRIPT
1085
URL
1086
S0360544213005525
RI PT
http://www.sciencedirect.com/science/article/pii/
1087
[32] E. Merkel, R. McKenna, W. Fichtner, Optimisation of the capacity and
1088
the dispatch of decentralised micro-chp systems: A case study for the uk,
1089
Applied Energy 140 (2015) 120–134. doi:10.1016/j.apenergy.2014.11.036.
1090
URL
1091
S0306261914011982
SC
http://www.sciencedirect.com/science/article/pii/
[33] M. Ameri, Z. Besharati, Optimal design and operation of district heating
1093
and cooling networks with CCHP systems in a residential complex, Energy
1094
and Buildings 110 (2016) 135–148. doi:10.1016/j.enbuild.2015.10.050.
1095
URL
1096
S0378778815303649
M AN U
1092
http://www.sciencedirect.com/science/article/pii/
1097
[34] P. Voll, C. Klaffke, M. Hennen, A. Bardow, Automated superstructure-
1098
based synthesis and optimization of distributed energy supply systems,
1099
Energy 50 (2013) 374–388. doi:10.1016/j.energy.2012.10.045.
1100
URL
1101
S0360544212008079
TE
D
http://www.sciencedirect.com/science/article/pii/
[35] E. Mehleri, H. Sarimveis, N. Markatos, L. Papageorgiou, A mathemat-
1103
ical programming approach for optimal design of distributed energy
1104
systems at the neighbourhood level, Energy 44 (1) (2012) 96–104.
1105
doi:10.1016/j.energy.2012.02.009.
1106
URL
1108
1109
1110
http://www.sciencedirect.com/science/article/pii/
AC C
1107
EP
1102
S036054421200103X
[36] E. Mehleri, H. Sarimveis, N. Markatos, L. Papageorgiou, Optimal design and operation of distributed energy systems:
Application
to Greek residential sector, Renewable Energy 51 (2013) 331–342.
1111
doi:10.1016/j.renene.2012.09.009.
1112
URL
1113
S0960148112005757
http://www.sciencedirect.com/science/article/pii/
47
ACCEPTED MANUSCRIPT
[37] H. Harb, J. Reinhardt, R. Streblow, D. M¨ uller, MIP approach for
1115
designing heating systems in residential buildings and neighbourhoods,
1116
Journal of Building Performance Simulation 9 (3) (2016) 316–330.
1117
doi:10.1080/19401493.2015.1051113.
1118
URL
1119
2015.1051113
RI PT
1114
http://www.tandfonline.com/doi/full/10.1080/19401493.
[38] M. A. Lozano, J. C. Ramos, L. M. Serra, Cost optimization of the design
1121
of CHCP (combined heat, cooling and power) systems under legal con-
1122
straints, Energy 35 (2) (2010) 794–805. doi:10.1016/j.energy.2009.08.022.
1123
URL
1124
S0360544209003624
SC
1120
M AN U
http://www.sciencedirect.com/science/article/pii/
1125
[39] D. Buoro, M. Casisi, P. Pinamonti, M. Reini, Optimal synthesis and
1126
operation of advanced energy supply systems for standard and do-
1127
motic home, Energy Conversion and Management 60 (2012) 96–105.
1128
doi:10.1016/j.enconman.2012.02.008.
1129
URL
1130
S0196890412000623 [40] R. Renaldi,
D
TE
1131
http://www.sciencedirect.com/science/article/pii/
A. Kiprakis,
D. Friedrich,
An optimisation frame-
1132
work for thermal energy storage integration in a residential heat
1133
pump heating system,
1134
doi:10.1016/j.apenergy.2016.02.067.
1135
URL
1136
S0306261916302045
1138
1139
EP
http://www.sciencedirect.com/science/article/pii/
AC C
1137
[41] T. of
their
Applied Energy 186 (3) (2017) 520–529.
Wakui,
residential operating
R.
Yokoyama,
Optimal
structural
design
cogeneration
systems
in
consideration
restrictions,
Energy
64
(2014)
of
719–733.
1140
doi:http://www.sciencedirect.com/science/article/pii/S0360544213008323.
1141
URL 10.1016/j.energy.2013.10.002
1142
[42] T. Wakui, H. Kawayoshi, R. Yokoyama, Optimal structural design of 48
ACCEPTED MANUSCRIPT
residential power and heat supply devices in consideration of operational
1144
and capital recovery constraints, Applied Energy 163 (2016) 118–133.
1145
doi:10.1016/j.apenergy.2015.10.154.
1146
URL
1147
S0306261915013963
RI PT
1143
http://www.sciencedirect.com/science/article/pii/
[43] H. D. Vinod, Integer Programming and the Theory of Grouping, Jour-
1149
nal of the American Statistical Association 64 (326) (1969) 506–519.
1150
doi:10.2307/2283635.
1151
URL http://www.jstor.org/stable/2283635 [44] F.
Dom´ınguez-Mu˜ noz, Gallardo-Salazar,
J.
M AN U
1152
SC
1148
M.
1153
M.
Selection
1154
CHP optimization,
1155
doi:10.1016/j.enbuild.2011.07.024.
1156
URL
1157
S037877881100329X
Cejudo-L´opez, of
typical
A.
Carrillo-Andr´es,
demand
days
for
Energy and Buildings 43 (2011) 3036–3043.
http://www.sciencedirect.com/science/article/pii/
[45] T. Sch¨ utz, M. Schraven, H. Harb, M. Fuchs, D. M¨ uller, Clustering algo-
1159
rithms for the selection of typical demand days for the optimal design of
1160
building energy systems, in: Proceedings of ECOS 2016 - The 29th In-
1161
ternational Conference on Efficiency, Cost, Optimization, Simulation and
1162
Environmental Impact of Energy Systems, University of Ljubljana, Faculty
1163
of Mechanical Engineering, 2016, pp. 1–12.
1165
1166
1167
1168
TE
EP
[46] Association of German Engineers, Economic efficiency of building instal-
AC C
1164
D
1158
lations - Fundamentals and economic calculation, Guideline VDI 2067-1, Beuth Verlag GmbH, D¨ usseldorf (Sep. 2012).
[47] T. Sch¨ utz, H. Harb, M. Fuchs, D. M¨ uller, Optimal Design of Building Energy Systems for Residential Buildings, in: P. K. Heiselberg (Ed.), CLIMA
1169
2016 - proceedings of the 12th REHVA World Congress, Vol. 6, Aalborg
1170
University, Department of Civil Engineering, 2016, pp. 1–10.
1171
URL http://vbn.aau.dk/files/233774153/paper_157.pdf 49
ACCEPTED MANUSCRIPT
[48] T. Sch¨ utz, L. Schiffer, H. Harb, M. Fuchs, D. M¨ uller, Optimal design of
1173
energy conversion units and envelopes for residential building retrofits
1174
using a comprehensive MILP model, Applied Energy 185 (2017) 1–15.
1175
doi:10.1016/j.apenergy.2016.10.049.
1176
URL
1177
S0306261916314933
http://www.sciencedirect.com/science/article/pii/
[49] Arbeitsgemeinschaft f¨ ur sparsamen und umweltfreundlichen Energiever-
SC
1178
RI PT
1172
brauch (ASUE), Bhkw-kenndaten 2014/15 (Oct. 2014).
1180
URL http://www.asue.de/blockheizkraftwerke/broschueren/05_10_
1181
14_bhkw-kenndaten_2014-15
1182
M AN U
1179
[50] Bundesnetzagentur, EEG-Umlage. Was ist die EEG-Umlage und wie
1183
funktioniert sie? (Oct. 2016).
1184
URL
1185
Sachgebiete/Energie/Verbraucher/Energielexikon/EEGUmlage.html
1188
& Sons, Inc., 2013.
D
1187
[51] H. P. Williams, Model Building in Mathematical Programming, John Wiley
[52] Bundesnetzagentur, Photovoltaikanlagen - Datenmeldungen und EEG-
TE
1186
https://www.bundesnetzagentur.de/SharedDocs/FAQs/DE/
1189
Verg¨ utungss¨ atze (May 2017).
1190
URL
1191
ElektrizitaetundGas/Unternehmen_Institutionen/
1192
ErneuerbareEnergien/Photovoltaik/DatenMeldgn_EEG-VergSaetze/
1193
DatenMeldgn_EEG-VergSaetze_node.html
1195
1196
1197
1198
EP
AC C
1194
https://www.bundesnetzagentur.de/DE/Sachgebiete/
[53] EEX, KWK Index (Jun. 2017). URL
https://www.eex.com/en/market-data/power/spot-market/
kwk-index
[54] J. A. Duffie, W. A. Beckman, Solar Engineering of Thermal Processes, 4th Edition, John Wiley & Sons, Inc., 2013.
50
ACCEPTED MANUSCRIPT
[55] R. Perez, P. Ineichen, R. Seals, J. Michalsky, R. Stewart, Modeling daylight
1200
availability and irradiance components from direct and global irradiance,
1201
Solar Energy 44 (1990) 271–289. doi:10.1016/0038-092X(90)90055-H.
1202
URL
1203
0038092X9090055H
1204
RI PT
1199
http://www.sciencedirect.com/science/article/pii/
[56] S. Dubey,
J. N. Sarvaiya,
B. Seshadri,
Temperature Dependent
Photovoltaic (PV) Efficiency and Its Effect on PV Production
1206
in the World - A Review, Energy Procedia 33 (2013) 311–321.
1207
doi:10.1016/j.egypro.2013.05.072.
1208
URL
1209
S1876610213000829
http://www.sciencedirect.com/science/article/pii/
M AN U
1210
SC
1205
[57] United Nations, Adoption of the Paris Agreement (2015).
1211
URL http://unfccc.int/resource/docs/2015/cop21/eng/l09r01.pdf
1212
[58] Umweltbundesamt, Entwicklung der spezifischen Kohlendioxid-Emissionen des deutschen Strommix in den Jahren 1990 - 2016 (May 2017).
1214
URL
1215
medien/1410/publikationen/2017-05-22_climate-change_15-2017_
1216
strommix.pdf
D
1213
TE
https://www.umweltbundesamt.de/sites/default/files/
[59] L. A. Wolsey, Integer Programming, Vol. 42, Wiley New York, 1998.
1218
[60] T. Wakui, R. Yokoyama, Optimal structural design of residential co-
1219
generation systems with battery based on improved solution method
1220
for mixed-integer linear programming, Energy 84 (2015) 106–120.
1222
1223
1224
AC C
1221
EP
1217
doi:10.1016/j.energy.2015.02.056. URL
http://www.sciencedirect.com/science/article/pii/
S036054421500211X
[61] Bundesamt
1225
vironment
1226
Aktualisierte
f¨ ur
Bauwesen
Consulting und
und
Potsdam
erweiterte
Raumordnung, GmbH,
Deutscher
Testreferenzjahre
51
Climate
von
&
En-
Wetterdienst, Deutschland
ACCEPTED MANUSCRIPT
1227
f¨ ur
mittlere,
extreme
und
zuk¨ unftige
Witterungsverh¨altnisse,
1228
URL
1229
Regelungen/Testreferenzjahre/Testreferenzjahre/01_start.html?
1230
nn=739044¬First=true&docId=743442 [accessed 24.01.2017] (Jul.
1231
2011).
RI PT
http://www.bbsr-energieeinsparung.de/EnEVPortal/DE/
[62] D. M¨ uller, M. Lauster, A. Constantin, M. Fuchs, P. Remmen, AIXLIB
1233
- AN OPEN-SOURCE MODELICA LIBRARY WITHIN THE IEA-EBC
1234
ANNEX 60 FRAMEWORK, in: BauSIM 2016, 2016, pp. 3–9.
1235
URL http://www.iea-annex60.org/downloads/2016-bausim-aixlib.
1236
pdf [63] P. Remmen,
M. Lauster,
M AN U
1237
SC
1232
M. Mans,
M. Fuchs,
T. Osterhage,
1238
D. M¨ uller, TEASER: an open tool for urban energy modelling of
1239
building stocks, Journal of Building Performance Simulation (2017)
1240
1–15doi:10.1080/19401493.2017.1283539.
1241
URL
1242
2017.1283539
D
[64] I. Richardson, M. Thomson, D. Infield, C. Clifford, Domestic electricity
TE
1243
http://www.tandfonline.com/doi/full/10.1080/19401493.
1244
use:
1245
42 (10) (2010) 1878–1887. doi:10.1016/j.enbuild.2010.05.023.
1246
URL
1247
S0378778810001854
1249
1250
1251
1252
EP
http://www.sciencedirect.com/science/article/pii/
[65] IEA Energy Conservation in Buildings & Community Systems, Annex 42
AC C
1248
A high-resolution energy demand model, Energy and Buildings
The Simulation of Building-Integrated Fuel Cell and Other Cogeneration Systems (COGEN-SIM), accessed May 11th 2016 (2008). URL http://www.ecbcs.org/annexes/annex42.htm
[66] M. Chen, G. A. Rinc´ on-Mora, Accurate electrical battery model capable
1253
of predicting runtime and i-v performance, IEEE Transactions on Energy
1254
Conversion 21 (2) (2006) 504–511. doi:10.1109/TEC.2006.874229.
1255
URL http://ieeexplore.ieee.org/abstract/document/1634598/ 52
ACCEPTED MANUSCRIPT
1257
1258
[67] STAWAG, accessed June 21st 2017 (2017). [link]. URL https://www.stawag.de/ [68] Umweltbundesamt,
Verhindern Herkunftsnachweise ”Greenwashing”?,
1259
accessed June 20th 2017 (Aug. 2013).
1260
URL
1261
verhindern-herkunftsnachweise-greenwashing
https://www.umweltbundesamt.de/service/uba-fragen/
¨ [69] Umweltbundesamt, Marktanalyse Okostrom (Apr. 2014).
SC
1262
RI PT
1256
1263
URL
1264
medien/376/publikationen/texte_04_2014_marktanalyse_
1265
oekostrom_0.pdf
M AN U
1266
https://www.umweltbundesamt.de/sites/default/files/
[70] MONTANA Energie-Handel, MONTANA Biogas, accessed June 21st 2017
1267
(Jun. 2017).
1268
URL
1269
montana-erdgas/tarife/biogas/
[71] Statistisches Bundesamt, Preise, daten zur energiepreisentwicklung (May
D
1270
https://www.montana-energie.de/privatkunden/erdgas/
1271
2017).
1272
URL
1273
Preise/Energiepreise/EnergiepreisentwicklungPDF_5619001.pdf?
1274
__blob=publicationFile
TE
EP
1275
https://www.destatis.de/DE/Publikationen/Thematisch/
[72] Gurobi Optimization, Linear Programming (LP) - A Primer on the Basics, accessed June 26th 2017 (2017).
1277
URL http://www.gurobi.com/resources/getting-started/lp-basics
1278
1279
1280
AC C
1276
[73] H. Wolisz, T. Sch¨ utz, T. Blanke, M. Hagenkamp, M. Kohrn, M. Wesseling, D. M¨ uller, Cost optimal sizing of smart buildings’ energy system components considering changing end-consumer electricity markets,
1281
Energydoi:10.1016/j.energy.2017.06.025.
1282
URL
1283
S0360544217310198
http://www.sciencedirect.com/science/article/pii/
53
ACCEPTED MANUSCRIPT
Table C.1: Variables and parameters
RI PT
Appendix C. Tables and Figures
Symbol
Description
Unit
A
Area
CRF
Capital Recovery Factor
E˙
Gas consumption
El
Annual el. consumption
Etar
Annual el. consumption limits of tariffs
MWh
G
Annual gas consumption of heating devices
MWh
Gtar
Annual gas consumption limits of tariffs
MWh
I
Solar irradiation
kW
M
Big-M, upper bound for a specific variable
—
Q˙
Heat flow rate
kW
P
Electrical power
kW
S
Storage’s state of charge
m2
a−1
M AN U
SC
kW
Temperature
V
Volume
TE
D
T
b
MWh
% ◦
C
m3
Price dynamic cash value
—
Costs
Euro
Storage capacity
kWh
Storage charging power
kW
Storage discharging power
kW
AC C e
Revenue
Euro
emi
Emissions
kg
f
Fixed parameters
—
k1
STC linear loss factors
W/(m2 ·K)
k2
STC quadratic loss factors
10−3 ·W/(m2 ·K2 )
p
Subsidy and remuneration rate
—
c cap ch dch
EP
1284
Continued on next page 54
ACCEPTED MANUSCRIPT
Table C.1: Variables and parameters
Description
rv
Residual value of each type of device
sub
Subsidies
t
Time period
tar
(Binary) decision on tariff selection
tax
Tax
w
Weighting variable
x
(Binary) decision of purchase
y
(Binary) activation of heating devices
z
Number of installed solar modules
Unit
RI PT
Symbol
—
Euro h
—
SC
—
—
—
M AN U
— —
Table C.2: Greek letters
Unit
Difference
—
TE
∆
PV loss factor
%/K
δ
(Binary) decision on KWKG subsidy
—
ζ
Linearized product of two variables
—
EP
γ
Description
D
Symbol
Efficiency
%
κ
Heat capacity
J / (kg·K)
ρ
Density
kg / m3
ϕ
Storage’s loss factor
—
AC C
η
55
ACCEPTED MANUSCRIPT
Table C.3: Subscripts and abbreviations
Description
BAT
Battery
BES
Building Energy System
BOI
Boiler
CFWH
Continuous flow water heater
CHP
Combined Heat and Power
COP
Coefficient of Performance
DHL
Design Heat Load
DHW
Domestic Hot Water
EEG
German Renewable Energy Sources Act
EEX
European Energy Exchange
EH
Electrical resistance Heater
EM
Electricity mix
SC
M AN U
HP
Heat pump
International Energy Agency
D
IEA INV
Inverter
German Reconstruction Credit Institute
TE
KfW
RI PT
Symbol
KWKG
German Act on Combined Heat and Power Generation
LB
Lower bound
Mixed-Integer Linear Program
MINLP
Mixed-Integer Nonlinear Program
PL
House’s plug loads
AC C
EP
MILP
PV
Photovoltaic
RES
Renewable Energy System
SH
Space Heating
SOS2
Special Ordered Set of Type 2
STC
Solar Thermal Collector
Continued on next page
56
ACCEPTED MANUSCRIPT
Description
TES
Thermal Energy Storage
UB
Upper bound
VDI
Association of German Engineers
ann
Annualized
cap
Capacity
d
Typical demand day
dem
Demand
dev
Device
M AN U
eeg
EEG levy
el
Electricity
exp
Export
feed
Feed-in
fix
Fixed price Gas
D
gas hpt
Heat pump tariff
Counting index, type index
TE
i ∗
i
Set of small scale micro CHP units
i∗
Set of large scale micro CHP units Import
infl
Inflation
inv
Investment
AC C
EP
imp
j
j
SC
Symbol
RI PT
Table C.3: Subscripts and abbreviations
∗
Tariff index Subset of non-hpt electricity tariffs
k
Counting index, data sheet interpolation
lvl
Tariff level
max
Maximum
Continued on next page
57
ACCEPTED MANUSCRIPT
Description
mds
Manufacturer data sheet
met
Metering
nom
Nominal
o&m
Operation and maintenance
sell
Sold to the distribution grid
spec
Specific
sub
Subsidies
var
Variable price
M AN U
SC
Symbol
RI PT
Table C.3: Subscripts and abbreviations
Table C.4: Summary of considered German subsidies and market characteristics
Implication
EEG (feed-in limit)
Max. 70% of PV peak power can be fed into the grid
46-49
EEG (levy)
40% of the EEG levy has to be paid on self-consumed electricity
18-24
D
Market characteristic / Subsidy
Equations
EEG (feed-in remuneration)
TE
This only applies to facilities with more than 10 kW capacity Special feed-in remuneration for electricity from PV
25-29
EP
Special rates for different installed capacities
KWKG
Financial support for battery assisted PV systems
38-42
However, max. 50% of PV peak power can be fed into the grid
46-49
Variable payment for self-consumption and feed-in from CHPs
31-37
AC C
KfW 275
Alternatively fixed payment for small-scale units
EStG HP tariffs
Gas and electricity tariffs
Fuel tax exemption for CHP units
9
Option for cheaper electricity tariff for HPs
11
Multiple tariffs with different fixed and variable costs
2-17
Tiered pricing structure Different emission factors for each tariff
58
67-68
RI PT
ACCEPTED MANUSCRIPT
Table C.5: Summary of investigated buildings
Type
Location
Apartments
Residents
Floor area in m
2
Roof area in m
SFH
Bavaria
1
3
121
40
2
MFH
Hamburg
8
19
520
100
3
AB
Berlin
15
34
1005
170
SH demand
Electricity demand
in kWh/a
in kWh/a
in kWh/a
950
6050
3750
8550
26035
18700
15450
50315
33950
M AN U
1
DHW demand
2
SC
Number
Type
BOI
1
E˙ mds
cinv
in kW
in kW
in kW
in Euro
—
2.8 ; 4.2 ; 15.8
1,970
4.1 ; 6.1 ; 20.3
—
4.1 ; 6.2 ; 23.2
2,120
3
5.5 ; 8.3 ; 27.7
—
5.6 ; 8.5 ; 31.7
2,305
4
7.4 ; 11.0 ; 36.8
—
7.4 ; 11.3 ; 42.1
2,680
1
2.5
1.0
4.2
9,585
2
4.7 ; 8.0
1.5 ; 3.0
7.7 ; 13.3
15,853
3
4.7 ; 12.5
1.5 ; 4.7
7.7 ; 21.1
19,472
4
12.3 ; 16.1 ; 20.1
4.3 ; 6.4 ; 8.5
20.3 ; 26.4 ; 33.5
25,542
5
24.0 ; 36.0 ; 45.0
10.0 ; 15.0 ; 20.0
45.5 ; 63.3 ; 79.9
38,002
EP
AC C
P mds
2.8 ; 4.1 ; 13.8
2
CHP
Q˙ mds
TE
Component
D
Table C.6: Available gas-fired heat generators.
59
ACCEPTED MANUSCRIPT
P nom
cinv
in kW
in kW
in Euro
CFWH / EH
1
2.0
2.0
179
2
6.0
6.0
199
3
8.0
8.0
4
12.0
12.0
HP
1
5.0
1.3
2 3
209
219
4,390
8.0
2.1
4,990
11.0
2.9
6,548
15.0
3.9
7,460
D
4
SC
Q˙ nom
Type
M AN U
Component
RI PT
Table C.7: Available electrical heat generators.
Table C.8: Scaled characteristics of the HP units.
(Scaled) Q˙ mds / (scaled) P mds
in ◦ C
—
TE
Ambient temperature
0.32 / 0.17
0.32 / 0.17
0.09 / 0.04
-15
0.57 / 0.27
0.51 / 0.22
0.12 / 0.05
-7
0.61 / 0.26
0.55 / 0.22
0.13 / 0.05
-3
0.69 / 0.26
0.62 / 0.22
0.14 / 0.05
0
0.74 / 0.26
0.66 / 0.22
0.15 / 0.05
2
0.79 / 0.26
0.70 / 0.23
0.16 / 0.05
7
1.08 / 0.28
0.96 / 0.23
0.21 / 0.05
10
1.17 / 0.28
1.05 / 0.24
0.23 / 0.05
20
1.48 / 0.27
1.33 / 0.23
0.29 / 0.05
AC C
EP
-20
60
Table C.9: Available solar thermal collectors.
Type
RI PT
ACCEPTED MANUSCRIPT
cinv
A
η0
k1
k2
in m2
—
in W/(m2 ·K)
in 10−3 ·W/(m2 ·K2 )
2.00
0.80
3.24
11.7
2
0.90
0.73
3.47
8.0
3
2.32
0.82
3.33
440
SC
1
in Euro
M AN U
23.0
Table C.10: Available solar power generators and inverters.
AC C
η
γ
cinv
in m2
in kW
—
%/K
in Euro
1.73
0.36
0.21
0.30
539
2
1.71
0.30
0.17
0.41
224
3
1.64
0.27
0.16
0.42
184
1
—
2.4
0.98
—
703
2
—
5.1
0.97
—
1,040
3
—
8.8
0.97
—
1,358
4
—
10.0
0.98
—
1,491
5
—
15.6
0.98
—
1,661
6
—
21.2
0.98
—
1,812
7
—
28.0
0.98
—
2,563
8
—
36.0
0.98
—
3,716
EP
INV
1
P nom
A
D
PV
Type
TE
Component
61
190 299
SC
RI PT
ACCEPTED MANUSCRIPT
Table C.11: Available storage units.
Type
cap
dch,max PBAT
cinv
in kWh/24h
—
kW
kW
in Euro
1
6.4
—
0.96
2.0
2.0
5,098
2
8.0
—
0.96
3.0
3.0
5,620
3
11.0
—
0.96
3.7
3.7
6,679
4
16.0
—
0.96
3.7
3.7
7,979
5
22.0
—
0.96
3.7
3.7
9,129
1
0.12
1.7
—
—
—
389
0.49
2.2
—
—
—
650
0.75
4.5
—
—
—
769
0.98
3.2
—
—
—
980
5
1.50
4.1
—
—
—
1,380
6
2.00
4.6
—
—
—
1,840
2 3
AC C
EP
4
TE
TES
ch,max PBAT
D
in kWh | m3 BAT
η
ϕ
M AN U
Component
62
RI PT
ACCEPTED MANUSCRIPT
EtarUB
cvar
cfix
espec
in MWh/a
in MWh/a
in Euro/kWh
in Euro/kWh
in kg/kWh
Standard el.
1
0.00
2.80
0.2713
73.02
0.527
2
2.80
6.00
0.2699
77.02
0.527
3
6.00
100.00
0.2687
84.16
0.527
Eco el.
1
0.00
2.80
0.2802
73.02
0.332
2
2.80
6.00
0.2735
92.02
0.332
3
6.00
9.00
0.2723
99.16
0.332
4
9.00
12.00
0.2713
107.73
0.332
12.00
100.00
0.2705
117.73
0.332
0.00
100.00
0.2013
91.51
0.527
1
0.00
5.38
0.0798
39.98
0.250
2
5.38
12.28
0.0615
138.66
0.250
3
12.28
100.00
0.0580
182.50
0.250
1
0.00
5.38
0.0858
39.98
0.225
2
5.38
12.28
0.0675
138.66
0.225
3
12.28
100.00
0.0640
182.50
0.225
5
AC C
Eco gas
EP
Standard gas
1
TE
Heat pump el.
M AN U
EtarLB
Level
D
Tariff
SC
Table C.12: Available electricity and gas tariffs.
63
ACCEPTED MANUSCRIPT
Value
Unit
10
a
5
%
Price change electricity
1.0512
1/a
Price change gas
1.0074
1/a
Price change EEX compensation
0.9551
1/a
15
a
Observation period Interest rate
M AN U
Lifetime CHP Lifetime boiler Lifetime EH Lifetime CFWH Lifetime HP Lifetime PV Lifetime STC
D
Lifetime TES Lifetime BAT
SC
Parameter
RI PT
Table C.13: Economic parameters.
a
20
a
20
a
18
a
20
a
20
a
20
a
15
a
15
a
EP
TE
Lifetime INV
20
AC C
Table C.14: Model constraints and variables.
New model
Original model
No subsidies
21993
16747
21948
1560
1560
1560
26134
24285
26108
Number integer variables
9
9
9
Number binary variables
2952
2943
2942
Number constrains
Number SOS constraints Number variables
64
RI PT
ACCEPTED MANUSCRIPT
Cost minimization Original
No subsidies
New model
M AN U
New model
SC
Table C.15: Results, single family house
Opt.
Rec.
20% CO2 reduction Original
No subsidies
Opt.
Rec.
1879.75
1870.60
1877.94
2158.01
1879.75
1870.60
1877.94
2165.21
Total emissions in tCO2 /a
-0.100
-0.078
-0.099
2.921
-0.100
-0.078
-0.099
2.248
Total investments in Euro/a
781.46
768.80
779.58
409.51
781.46
768.80
779.58
530.29
TES investments in Euro/a
58.34
45.68
56.46
34.91
58.34
45.68
56.46
69.20
Demand costs in Euro/a
1459.14
1464.80
1459.53
1488.57
1459.14
1464.80
1459.53
1476.35
Metering costs in Euro/a
215.68
215.68
215.68
215.68
215.68
215.68
215.68
117.00
Revenues in Euro/a
670.66
670.66
670.66
19.55
670.66
670.66
670.66
37.79
BOI in kW
13.8
EH in kW
0
3
0.49
PV in m2
37.62
2
STC in m
TE
13.8
13.8
13.8
13.8
13.8
13.8
0
0
2
0
0
0
2
0.33
0.49
0.12
0.49
0.33
0.49
0.75
37.62
37.62
8.18
37.62
37.62
37.62
11.45
2.32
2.32
2.32
2.32
2.32
2.32
2.32
6.96
660.8
1340.7
203.4
753.3
606.2
1340.7
203.4
851.1
AC C
Run times in s
13.8
EP
TES in m
D
Total costs in Euro/a
65
RI PT
ACCEPTED MANUSCRIPT
Table C.16: Results, multi family house
Cost minimization Original
No subsidies
Opt.
Rec.
No subsidies
Opt.
Rec.
7313.96
7318.92
7818.05
7229.53
7214.07
7216.87
Total emissions in tCO2 /a
12.449
12.525
12.528
11.494
8.964
9.036
9.042
9.184
Total investments in Euro/a
3892.94
3891.7
3895.31
2411.75
4322.08
4318.31
4324.45
3951.58
TES investments in Euro/a
87.96
86.72
90.33
69.20
87.96
84.19
90.33
69.20
Demand costs in Euro/a
4105.34
4106.32
4105.66
4645.83
3854.36
3850.29
3849.19
3556.4
Metering costs in Euro/a
259.52
255.52
255.52
266.66
255.52
255.52
255.52
259.52
EEG levy in Euro/a
0
0
0
0
459.83
462.09
462.41
0
Revenues in Euro/a
675.95
681.16
682.08
24.74
1245.61
1245.06
1246.96
175.85
Subsidies in Euro/a
440.92
446.79
446.63
0
445.40
451.54
451.10
0
BOI in kW
0
CHP in kW
12.5
EH in kW
12
TE
D
M AN U
7325.72
Original
Total costs in Euro/a
0
0
20.3
0
0
0
20.3
12.5
12.5
0
12.5
12.5
12.5
2.5
12
12
8
12
12
12
0
EP
BAT in kWh
7526.96
New model
SC
New model
20% CO2 reduction
0
0
0
22
0
0
0
22
0.98
0.93
0.98
0.75
0.98
0.89
0.98
0.75
PV in m2
60.52
60.52
60.52
99.78
99.78
99.78
99.78
89.97
2
0
0
0
0
0
0
0
9.28
Run times in s
2935.7
1969.8
183.2
1449.9
1581.0
710.5
32.0
1777.6
3
AC C
TES in m
STC in m
66
Table C.17: Results, apartment building
Cost minimization No subsidies
Opt.
Rec.
New model
SC
Original
20% CO2 reduction Original
No subsidies
Opt.
Rec.
11132.50
11120.01
11127.67
11715.65
11400.85
11405.32
11415.20
11774.60
Total emissions in tCO2 /a
15.432
15.547
15.566
21.016
12.346
12.438
12.438
16.813
Total investments in Euro/a
5305.27
5297.73
5307.69
5334.66
6304.78
6300.52
6300.47
5808.07
TES investments in Euro/a
123.86
116.32
126.28
87.96
165.14
150.56
160.84
87.96
Demand costs in Euro/a
8338.83
8347.89
8344.1
6046.94
7552.32
7496.45
7605.69
5809.54
Metering costs in Euro/a
266.66
266.66
266.66
259.52
274.52
274.52
274.52
255.52
EEG levy in Euro/a
747.07
750.48
751.46
0
794.60
795.07
796.87
0
Revenues in Euro/a
2777.07
2784.86
2784.86
189.53
2656.01
2607.03
2656.06
401.95
Subsidies in Euro/a
986.93
995.38
996.58
0
1118.63
1102.81
1159.99
0
BOI in kW
13.8
13.8
13.8
20.3
13.8
13.8
13.8
20.3
CHP in kW
12.5
12.5
12.5
8
20.1
20.1
20.1
8
EH in kW
12
12
12
12
6
6
6
12
BAT in kWh
0
0
0
22
0
0
0
22
1.50
1.36
1.50
0.98
2.00
1.85
2.00
0.98
168.49
168.49
168.49
127.59
166.85
165.22
166.85
166.85
PV in m2
D
TE
EP
TES in m3
M AN U
Total costs in Euro/a
AC C
New model
RI PT
ACCEPTED MANUSCRIPT
STC in m2
0
0
0
2.32
2.32
4.64
2.32
2.32
El. tariff
std.
std.
std.
std.
eco
eco
eco
std.
Run times in s
3283.0
3136.1
0.4
528.4
4613.0
4842.2
78.3
1275.5
67
Table C.18: Sensitivity analysis
Opt.
+5% demands
Rec.
Opt.
-5% tariffs
Rec.
Single family house
Opt.
SC
-5% demands
RI PT
ACCEPTED MANUSCRIPT
+5% tariffs
Rec.
Opt.
Rec.
1802.71
1811.01
1959.11
1959.02
1806.83
1806.69
1952.88
1952.60
BOI in kW
13.8
13.8
13.8
13.8
13.8
13.8
13.8
13.8
EH in kW
2
2
0
0
0
0
0
0
TES in m3
0.12
0.49
0.49
0.49
0.49
0.49
0.49
0.49
M AN U
Total costs in Euro/a
Multi family house 6904.51
7007.99
7453.44
7452.74
6959.55
7007.18
7450.71
7450.40
BOI in kW
27.7
0
0
0
27.7
0
0
0
CHP in kW
0
12.5
12.5
12.5
0
12.5
12.5
12.5
EH in kW
0
12
12
12
0
12
12
12
3
0.49
0.98
0.98
0.98
0.49
0.98
0.98
0.98
PV in m2
93.24
60.52
60.52
60.52
94.88
60.52
60.52
60.52
2
4.64
0
0
0
4.64
0
0
0
TE
STC in m
EP
TES in m
D
Total costs in Euro/a
Apartment building
10612.79
10609.87
11631.12
11688.43
10688.52
10683.92
11582.37
11582.68
BOI in kW
13.8
13.8
20.3
13.8
13.8
13.8
13.8
13.8
AC C
Total costs in Euro/a
CHP in kW
12.5
12.5
20.1
12.5
12.5
12.5
12.5
12.5
EH in kW
12
12
0
12
12
12
12
12
3
1.50
1.50
2.00
1.50
1.50
1.50
1.50
1.50
TES in m
68
5
RI PT
Continuous Discrete
SC
40 35 30 25 20 15 10 5 0 0
10
15
M AN U
Investment costs in 1000 EUR
ACCEPTED MANUSCRIPT
Nominal electricity output in kW
20
Figure C.1: Continuous and discrete approaches for CHP units (single column figure)
15 10
D TE
20
Constant efficiency No part load threshold Piecewise linearization
EP
Electrical efficiency in %
25
5
AC C
0 0
20
40
60
Modulation level in %
80
100
Figure C.2: Part load models for one exemplary CHP unit (single column figure)
69
−
Electrical loads
DHW CFWH
Boiler CHP EH HP
−
TES
BAT
+
M AN U
Natural gas
SC
SH
~
RI PT
ACCEPTED MANUSCRIPT
Electrical grid
AC C
EP
TE
D
Figure C.3: Building energy system structure (single column figure)
Figure C.4: Building’s heat balance (single column figure)
70
EP
TE
D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
Figure C.5: Building’s electricity balance (single column figure)
71
ACCEPTED MANUSCRIPT Table C.4: Summary of considered German subsidies and market characteristics
EEG (levy) EEG (feed-in remuneration) KfW 275
KWKG
Max. 70% of PV peak power can be fed into the grid 40% of the EEG levy has to be paid on self-consumed electricity This only applies to facilities with more than 10 kW capacity Special feed-in remuneration for electricity from PV Special rates for different installed capacities Financial support for battery assisted PV systems However, max. 50% of PV peak power can be fed into the grid Variable payment for self-consumption and feed-in from CHPs Alternatively fixed payment for small-scale units Fuel tax exemption for CHP units Option for cheaper electricity tariff for HPs Multiple tariffs with different fixed and variable costs Tiered pricing structure Different emission factors for each tariff
Equations 46-49 18-24
25-29 38-42 46-49 31-37
Citation in the text (lines 237-239):
9 11 2-17 67-68
M AN U
SC
EStG HP tariffs Gas and electricity tariffs
Implication
RI PT
Market characteristic / Subsidy EEG (feed-in limit)
Latex version:
TE D
A main novelty of this paper is the consideration of many German subsidies and market characteristics. Table C.4 summarizes the implemented characteristics and links them to the corresponding equations used in this paper.
AC C
EP
\begin{table}[h!] \caption{Summary of considered German subsidies and market characteristics} \centering \hspace*{-10em} \begin{tabular}[l]{@{}lll} \hline Market characteristic / Subsidy & Implication & Equations\\ \hline EEG (feed-in limit) & Max. 70\% of PV peak power can be fed into the grid & \ref{eqn:limit feed in}-\ref{eqn:end_limit_feed_in}\\ EEG (levy) & 40\% of the EEG levy has to be paid on self-consumed electricity & \ref{eqn: eeg more than 10 kW installed_1}-\ref{eqn:eeg_levy_end}\\ & This only applies to facilities with more than 10 kW capacity & \\ EEG (feed-in remuneration) & Special feed-in remuneration for electricity from PV & \ref{eqn:rev_feed_in_pv_1}-\ref{eqn:rev_feed_in_pv_ende}\\ & Special rates for different installed capacities & \\ \hline KfW 275 & Financial support for battery assisted PV systems & \ref{eqn: sub_bat_start}-\ref{eqn: sub_bat_end}\\ & However, max. 50\% of PV peak power can be fed into the grid & \ref{eqn:limit feed in}-\ref{eqn:end_limit_feed_in}\\ \hline
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
KWKG & Variable payment for self-consumption and feed-in from CHPs & \ref{eqn:subs_CHP_large}-\ref{eqn:kwkg_ende} \\ & Alternatively fixed payment for small-scale units & \\ \hline EStG & Fuel tax exemption for CHP units & \ref{eqn:cost_gas_energy_tax_chp}\\ \hline HP tariffs & Option for cheaper electricity tariff for HPs & \ref{eqn: allow_hp_tariff} \\ \hline Gas and electricity tariffs & Multiple tariffs with different fixed and variable costs & \ref{eqn:tariff_start}-\ref{eqn:tariff_metering} \\ & Tiered pricing structure & \\ & Different emission factors for each tariff & \ref{eqn:co2_gas}-\ref{eqn:co2_el}\\ \hline \end{tabular} \label{tab: market characteristics} \end{table}
ACCEPTED MANUSCRIPT Highlights:
EP
TE D
M AN U
SC
RI PT
MILP with discrete, device-specific storage modeling Consideration of German subsidies, market characteristics and multiple tariffs Storage model is more accurate but marginally affects the calculating times Subsidies and market characteristics strongly influence the optimal energy systems Results are robust to variations in energy tariff costs and demands
AC C
1. 2. 3. 4. 5.
S0360-5442(17)31400-7
DOI:
10.1016/j.energy.2017.08.024
Reference:
EGY 11393
To appear in:
Energy
Received Date: 23 December 2016 Revised Date:
3 July 2017
Accepted Date: 6 August 2017
Please cite this article as: Schütz T, Schraven M, Remy S, Granacher J, Kemetmüller D, Fuchs M, Müller D, Optimal design of energy conversion units for residential buildings considering German market conditions, Energy (2017), doi: 10.1016/j.energy.2017.08.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
RI PT
Optimal design of energy conversion units for residential buildings considering German market conditions Thomas Sch¨ utz∗, Markus Schraven, Sebastian Remy, Julia Granacher, Dominik Kemetm¨ uller, Marcus Fuchs, Dirk M¨ uller
SC
RWTH Aachen University, E.ON Energy Research Center, Institute for Energy Efficient Buildings and Indoor Climate, Mathieustr. 10, Aachen, Germany
Abstract
M AN U
Many countries have passed governmental action plans to support the installation of renewable energy sources. However, most studies dealing with the optimization of building energy systems neglect a precise modeling of such subsidies, although these subsidies are specifically designed to strongly influence system setups. Therefore, this paper extends a model for the optimization of energy systems by a more accurate consideration of storage units and enhance
D
both models by accounting for major German pieces of legislation aimed at supporting renewable energies. Additionally, we consider typical German mar-
TE
ket characteristics, in particular the availability of multiple gas and electricity tariffs.
EP
We compare our model with the original formulation regarding a pure cost minimization and a forced reduction of CO2 emissions for three new buildings located in Germany. The results imply that the considered subsidies strongly
AC C
support the installation of PV modules and CHP units. Without these subsidies, batteries and solar thermal collectors become more important. Additionally, the findings illustrate that the new storage model is slightly more accurate, but only marginally affects the total annual costs and required computing times. The conducted sensitivity analysis has shown that the obtained results are relatively ∗ Corresponding
author Email address: [email protected] (Thomas Sch¨ utz)
Preprint submitted to Energy
August 7, 2017
ACCEPTED MANUSCRIPT
robust to variations in energy tariff costs and demands. Keywords: Building Energy Systems, German regulations, Mixed-Integer
1
RI PT
Linear Programming, Multiple energy tariffs
1. Introduction
The transition towards a more energy efficient economy with lower CO2
3
emissions is a recognized objective of the European Union [1]. In Germany, this
4
concept is known as “Energiewende” and aims at reducing greenhouse gas emis-
5
sions, increasing electricity generation from Renewable Energy Sources (RES)
6
and achieving higher energy efficiency in general [2]. In the context of buildings,
7
which account for approx. 40% of total energy consumption in the European
8
Union [1], emission reductions and energy savings can for example be achieved
9
by installing more efficient heating devices and by improving their control strat-
M AN U
10
SC
2
egy.
In recent years, many different heat and electricity generation as well as
12
storage technologies evolved for application in buildings. Small-scale Combined
13
Heat and Power (CHP) units offer a highly efficient method for generating heat
14
and electricity simultaneously from fossil fuels [3, 4]. Potential benefits can
15
further be leveraged by introduction of Thermal Energy Storage (TES) devices
16
[5]. Heat Pump (HP) systems present a technology for efficiently using electricity
17
for heating purposes [6]. RES, especially solar systems can further be used on
18
building level, such as Solar Thermal Collectors (STCs) [7] or Photovoltaic (PV)
19
modules [8]. The integration of such fluctuating generators can for example be
21
22
23
TE
EP
AC C
20
D
11
enhanced by storage devices such as TES [9] and batteries (BATs) [10]. In order to achieve the proposed emission reductions and energy savings, the
German government supports the utilization of technologies such as RES and CHP in the building sector. For example, the Renewable Energy Sources Act
24
(German abbreviation EEG) [11], guarantees above market and long-term feed-
25
in tariffs for PV plants. Additionally, the Act on Combined Heat and Power
26
Generation (German abbreviation KWKG) [12] provides subsidies for feed-in
2
ACCEPTED MANUSCRIPT
and self-consumed electricity from CHPs. Further methods for promoting RES
28
and low CO2 technologies include subsidies from the German Reconstruction
29
Credit Institute (German abbreviation KfW) for combined PV and BAT systems
30
[13], tax exemptions for CHP units [14] as well as private utility companies
31
offering reduced electricity tariffs for HP systems.
32
1.1. Literature review
RI PT
27
In order to evaluate the economic suitability of a small number of predefined
34
Building Energy Systems (BESs), multiple, simplified simulation models have
35
been developed [15, 16, 17].
M AN U
SC
33
However, the vast quantity of combinations of available devices into a BES
37
and the specific subsidies require a systematic analysis and evaluation method
38
[18], making optimization approaches a viable option for determining the op-
39
timal structure, sizing and operation of BES. There are already optimization
40
frameworks available for energy system optimization, such as TIMES [19], DER-
41
CAM [20], and COMPOSE [21, 22]. These models may contain shortcom-
42
ings, such as oversimplifications leading to effects like the technology mix effect
43
[23] that can hardly be altered in such given frameworks. Therefore, many
44
researchers develop their own models for optimizing BES design, sizing, and
45
operation.
TE
D
36
The inherent nonlinearities arising in such optimization models, like the
47
typically nonlinear part load behavior of generation units, has led to the devel-
48
opment of accurate mixed-integer nonlinear programs (MINLP) [24, 25]. Ac-
49
cording to Klatt and Marquardt [26], however, MINLP is still not a suitable
51
52
53
AC C
50
EP
46
method for reasonably sized models. Therefore, MINLP models are often reformulated and solved using mixed-integer linear programming (MILP) or handled with heuristics [27, 28]. However, using heuristics has proven to be very time consuming and does not necessarily lead to significant improvements regarding
54
the obtained accuracy, if the model can be linearized properly [29]. Addition-
55
ally, MILP models can further be used as a good initial solution for a MINLP
56
solver [30]. 3
ACCEPTED MANUSCRIPT
Therefore, the majority of analyses dealing with the optimal design and
58
operation of BES use MILP, often requiring major simplifications. These sim-
59
plifications most frequently occur regarding the devices’ capacities and their
60
part load behavior. Device capacity can either be modeled in a continuous
61
manner or discretely, as illustrated in Figure C.1 for all CHP units considered
62
in this paper. Discrete modeling allows for assigning specific investment costs,
63
efficiency curves, nominal heat outputs, operating times, etc. to each device
64
individually, whereas in a continuous device modeling, a representative device
65
is scaled between lower and upper limits for the nominal heat or electricity out-
66
put. In the continuous model, all part load efficiency curves are the same for
67
one type of device (e.g. for all CHP units) and costs are typically based on lin-
68
ear regressions, as shown in Figure C.1. Additionally, the continuous approach
69
can lead to optimal results that are not available for purchase (e.g. the calcula-
70
tion requires a 15 kW CHP unit, whereas only 10 kW and 20 kW are available).
71
Therefore, the continuous approach presents a major simplification that reduces
72
the model’s accuracy, while typically improving the computing times.
M AN U
SC
RI PT
57
Furthermore, multiple approaches for modeling part load are available in
74
MILP models. Figure C.2 shows three different models for the electrical effi-
75
ciency of one CHP unit considered in this paper. The dashed blue line depicts
76
a model that neglects the switching on threshold and does not account for part
77
load deterioration. The red line shows a model that also does not consider part
78
load deterioration but accounts for a switching on threshold of the device. In
79
contrast, the black curve displays a piecewise linearization of the device’s part
80
load behavior, which has for example been considered by Pickering et al. [29].
82
83
84
TE
EP
AC C
81
D
73
The following paragraphs present studies dealing with the optimal design, sizing, and operation of energy systems using MILP and illustrate their underlying simplifications.
Ashouri et al. [31] presented a MILP framework for the optimal selection and
85
sizing of smart building systems in Switzerland, considering all aforementioned
86
generation and storage units, as well as chillers and ice storage systems. Their
87
device modeling considers continuous equipment sizes rather than available, 4
ACCEPTED MANUSCRIPT
discrete sizes. Furthermore, no switching on threshold or decreased part load
89
performance are modeled. Merkel et al. [32] used the same simplifications,
90
optimizing residential micro-CHP systems in the United Kingdom, considering
91
peak load boilers and hot water storage tanks.
RI PT
88
Other studies [33, 34] use this continuous dimensioning as done by Ashouri et
93
al. [31], however accounting for linear part load losses. Ameri and Besharati [33]
94
optimize district heating and cooling networks in Iran considering gas turbines,
95
boilers, chillers and PV. Voll et al. [34] compute an optimal energy system for
96
industrial applications in Germany comprising CHPs, boilers and chillers.
SC
92
Mehleri et al. [35, 36] model some equipment sizes, such as boilers, PV area
98
and TES volume with continuous variables, whereas CHP capacity is described
99
with discrete steps. They also neither account for switching on thresholds nor
100
part load deterioration. Their model is applied to optimize local neighborhoods
101
in Greece considering CHPs, boilers, PV, TES and district heating as well as
102
microgrids. They account for market characteristics by considering different
103
constant feed-in tariffs for electricity from CHP and PV.
M AN U
97
Such a mixed modeling of device capacities is also used by [37, 38]. Harb
105
et al. [37] model CHP and HP capacities discretely and rely on the continuous
106
sizing for PV, boilers and TES. In this model, CHP part load is handled with
107
an empirical approach, whereas constant efficiencies are assumed for boilers and
108
HP units for the entire modulation range. The model is applied to determine
109
optimal configurations for German residential buildings and extended to com-
110
pute local heating networks and microgrids for a small neighborhood. Harb
111
et al. [37] consider some aspects of German regulations, such as above-market
113
114
115
TE
EP
AC C
112
D
104
feed-in remuneration for electricity generated through PV as well as support for CHP units like tax exemptions for gas combusted in CHP units. However, they did not consider regulations impeding the operation and installation of specific energy conversion components, such as the EEG levy on self-consumption if the
116
installed capacity exceeds a certain threshold, a limit on the maximum feed-in of
117
PV as well as capacity specific remuneration for PV. Additionally, to the best of
118
our knowledge, this approach presents the only available model for considering 5
ACCEPTED MANUSCRIPT
two electricity tariffs during the design optimization, a special heat pump tariff
120
and a standard electricity tariff. Lozano et al. [38] present the optimization
121
of combined heat, cooling and power systems considering CHP, boilers, chillers
122
and TES for a city district in Spain. They model device capacities discretely
123
but represent TES sizes continuously. Their devices are assumed to be 2-point
124
controlled, not modeling part load. Furthermore, Lozano et al. [38] investigate
125
the effect of formerly required minimum self-consumption rates of electricity
126
generated through CHP units on the design and sizing of CCHP systems.
SC
RI PT
119
Buoro et al. [39] analyze fully automated homes in Italy by optimizing their
128
respective energy systems consisting of CHP, boiler, chiller, PV, STC and TES.
129
The part load is based on a linear regression without accounting for the devices’
130
activation threshold.
M AN U
127
Renaldi et al. [40] describe a framework for optimizing HP and TES sys-
132
tems for residential buildings in the United Kingdom. Their devices’ selection is
133
entirely based on discrete choices, however their TES model assumes constant
134
losses, even if the storage is totally discharged. Heat pumps’ temperature-
135
dependent COP is modeled, however the COP is assumed to be constant during
136
part load. Renaldi et al. [40] consider a governmental support for heat gener-
137
ation from HP with a linear relation between this revenue and the building’s
138
annual heat demand.
TE
D
131
Wakui and Yokoyama [41] present a model for optimizing energy systems
140
for Japanese residential buildings comprising CHP, TES, boiler and electrical
141
heaters (EHs). The selection of CHP and storage tanks is coupled, so that if
142
a certain CHP unit is chosen, a pre-specified storage tank is installed as well.
144
145
146
147
AC C
143
EP
139
Part load behavior is described by means of piecewise linearization, introducing multiple linear relationships that model the nonlinear part load curves. Wakui et al. [42] extended their model by further considering heat pumps. The selection of TES units has been decoupled from the selection of CHP units in this work; however, TES’ sizes are chosen via continuous variables rather than discretely.
148
In conclusion, the model developed by Wakui et al. [42] overcomes most
149
of the described simplifications applied in MILP models. We therefore used 6
ACCEPTED MANUSCRIPT
this model as a foundation for this work. Additionally, the literature review
151
shows that modeling of subsidies and market characteristics has largely been
152
overlooked or treated in a simplified manner in previous works. Remuneration
153
for electricity or heat has commonly been coupled linearly to the generation,
154
for instance neglecting decreased remuneration for large facilities. Additionally,
155
regulations impeding the operation of specific devices, such as feed-in limits or
156
costs for self-consumption have not been taken into account thoroughly.
157
1.2. Contributions
SC
RI PT
150
In order to contribute to the field of optimal design of building energy sys-
159
tems, we extend the model proposed by Wakui et al. [42] regarding device se-
160
lection and the implementation of German governmental subsidies and market
161
characteristics. Our work therefore presents the following three novelties.
M AN U
158
First, we extend the original model formulation to also select storage units
163
discretely. In this way, we assure that the determined, optimal system is avail-
164
able for purchase in reality, and we are able to include detailed information such
165
as standby losses, charging and discharging characteristics as well as investment
166
costs for each type of storage unit. Such discrete modeling has partly been done
167
by the studies cited above; however, to the best of our knowledge, TES have
168
not been accurately described in a discrete manner.
TE
D
162
The second novelty is a detailed modeling of many specific German regula-
170
tions and market characteristics, of which some support and others impede the
171
installation and operation of RES. Based on the current EEG, we account for
172
limitations on PV feed-in. We consider variable feed-in remuneration for PV,
174
175
176
AC C
173
EP
169
depending on the installed capacity as well as the EEG levy for self-consumed electricity if the installed peak generation capacity exceeds 10 kW. This paper models the KWKG considering remuneration for feed-in and self-consumption for electricity generated with CHP. Furthermore, the presented approach models
177
tax exemptions for CHP units’ fuel. Also, we consider subsidies for PV systems
178
with batteries according to KfW 275. Additionally, we account for reduced heat
179
pump tariffs that are often offered by utility companies in Germany. Since most 7
ACCEPTED MANUSCRIPT
of the previously mentioned papers use generic settings, the consideration of
181
local market characteristics improves the applicability of the model for realistic
182
investment decisions. In order to facilitate the transfer of this paper’s findings
183
to incentive programs and market characteristics in other countries, we present
184
the modeling approach in general formulations and detail the distinct features
185
of the German application.
RI PT
180
Our third contribution is the detailed modeling of multiple gas and electricity
187
tariffs. Previous studies only accounted for a single gas and electricity rate, or
188
in the case of Harb et at. [37] a single gas tariff and two electricity tariffs,
189
whereas we include the possibility of considering an arbitrary number of gas
190
and electricity tariffs and we also account for a tiered pricing structure of each
191
tariff. In this way, we allow for analyzing the trade-off between monetary costs
192
and environmental impacts.
M AN U
SC
186
These novelties are of manifold importance: Researchers benefit from a de-
194
tailed decision modeling. Regulators can investigate the effect of certain laws
195
on rational decisions. Practitioners can use the framework for determining op-
196
timal BES configurations and analyze trade-offs between economic and ecologic
197
objectives.
TE
D
193
The rest of this paper is structured as follows: Section 2 describes the devel-
199
oped optimization model. Afterwards, Section 3 presents the inputs for analyz-
200
ing our model. In order to evaluate our model extensions, we first compare our
201
developed model with the original formulation in Section 4. This comparison
202
is conducted for three new German residential buildings significantly differing
203
in size and location. Furthermore, we evaluate the importance of considering
205
206
207
208
AC C
204
EP
198
governmental subsidies and market characteristics on BES design, sizing and operation as well as the estimation of total costs. We also perform a sensitivity analysis for analyzing the robustness of the calculated solutions and discuss the weaknesses of our model. Finally, the findings are summarized and an outlook for future research is given in Section 5.
209
The model and calculations described in this paper can be downloaded
210
from https://github.com/RWTH-EBC/BESopt. (The currently private repos8
ACCEPTED MANUSCRIPT
itory will be made public upon acceptance of this paper.)
212
2. Modeling
RI PT
211
The structure of the considered building energy system is shown in Fig-
214
ure C.3. Conventional heat generators such as gas boilers, CHP units, EHs,
215
continuous flow water heaters (CFWHs) and electrical air/water HPs are con-
216
sidered. Furthermore, storage devices like BATs and TES units are available.
217
Solar generators like PV modules and STCs as well as peripheral devices like
218
inverters (INVs) are also included as possible parts of the optimal BES. Ther-
219
mal loads include space heating (SH) as well as domestic hot water (DHW).
220
Electrical loads describe the building’s electricity consumption for lighting and
221
electrical appliances like computers, refrigerators and televisions that we cur-
222
rently consider unshiftable. The implemented energy balances are described in
223
this chapter and summarized graphically in Appendix A.
M AN U
SC
213
The model requires annual inputs for thermal and electrical loads. Further
225
inputs include device-specific data, such as the available types (e.g. CHP type 1,
226
CHP type 2, etc.) of each device (e.g. CHP, HP, etc.) and their characteristics,
227
for instance efficiency curves, expected life time, investment costs as well as
228
operation and maintenance costs. Also, information regarding different gas and
229
electricity tariffs, economic parameters like interest rate, tax rates, length of the
230
observation period and subsidy rates can be specified.
EP
TE
D
224
Since modeling an entire year requires long computing times, the inputs are
232
clustered into multiple representative periods that are weighted with weighting
233
234
235
236
AC C
231
variables wd . The clustering is based on the k-medoids method [43] and additionally rescales the cumulated inputs in order to preserve the annual energy demands [44]. This method has been shown to provide reliable results of high quality for energy system optimization purposes [45].
237
A main novelty of this paper is the consideration of many German subsidies
238
and market characteristics. Table C.4 summarizes the implemented character-
239
istics and links them to the corresponding equations used in this paper.
9
ACCEPTED MANUSCRIPT
240
The remainder of this section explains the objective function as well as the economic, technical and ecological modeling.
242
2.1. Objective function
RI PT
241
The original model of Wakui et al. [42] minimizes the annual primary energy
244
consumption. However, we believe that most decisions regarding energy supply
245
are economically driven. Thus, we chose to minimize annual costs cann . An-
246
nual costs are the sum of costs for investments cinv , operation and maintenance
247
co&m , demand related costs cdem , metering cmet and EEG-levy for self-consumed
248
electricity ceeg less the revenues generated from feed-ins efeed and subsidies esub .
M AN U
SC
243
min cann = cinv + co&m + cdem + cmet + ceeg − efeed − esub
(1)
The following subsections focus on the constraints that are new in this pa-
250
per. Additionally required constraints are briefly described and listed in Ap-
251
pendix B as well as this project’s open-source repository (https://github.
252
com/RWTH-EBC/BESopt).
253
2.2. Economic constraints
The economic modeling is based on the German engineering guideline VDI 2067
TE
254
D
249
[46], that has also been used in the authors’ previous publications [37, 45, 47, 48].
256
Investment costs are distributed into equal, annual payments by means of
257
the capital recovery factor CRF . For STC and PV, the investment costs are
258
determined by the number of modules and their specific costs, whereas for other
259
devices the unit’s costs can be used directly.
261
262
263
AC C
260
EP
255
Costs for operation and maintenance for all devices but CHP units are mod-
eled as a fixed percentage of the initial investment. The corresponding values can be found in [46]1 . For CHP units, ASUE [49] offers more detailed models
for operation and maintenance related costs that are based on device operation, 1 These
costs are based on a German engineering guideline that is currently in force and
are therefore considered to be suitable. However, the framework allows for defining accurate values for each device, if such data are available.
10
ACCEPTED MANUSCRIPT
assuming a proportionality between the generated amount of electricity and the
265
resulting costs.
266
2.2.1. Demand related costs
RI PT
264
The modeling of demand related costs is a novelty of this publication, since
268
an arbitrary number of gas and electricity tariffs can be considered. In contrast,
269
all cited previous publications only used one gas tariff and at most two electricity
270
tariffs.
SC
267
Demand related costs depend on the chosen gas or electricity tariff. If a CHP or boiler is installed, the binary decision variable xdev,i indicating if type i
M AN U
of device dev has been purchased, would equal 1. In this case, a gas tariff tarjgas has to be selected. Furthermore, at most one gas tariff can be chosen. X X tarjgas ≥ xdev,i ∀ dev ∈ {CHP, BOI} j
X
i
tarjgas ≤ 1
j
(2) (3)
As many utilities offer a tiered pricing structure depending on the annual
272
gas,lvl consumption, different tariff levels tarj,l are introduced. At most one level
273
can be valid and if the corresponding gas tariff is not chosen, all level variables
274
have to be 0:
TE
D
271
X
gas,lvl tarj,l = tarjgas
∀j
(4)
l
The correct level is bounded by a lower bound on the minimum annual
276
LB consumption Gtarj,l and an upper bound on the maximum annual consumption
277
UB Gtarj,l :
AC C
EP
275
gas,lvl gas,lvl LB CHP UB tarj,l · Gtarj,l ≤ GBOI ≤ tarj,l · Gtarj,l j,l + Gj,l
278
279
280
281
∀j, l
(5)
In this equation, GBOI and GCHP describe the annual gas consumption of j,l j,l
CHP units and boilers at tariff j and level l. For boilers and CHP units, the annual gas consumption Gann dev of all days d
and time steps t results in: X XX Gann wd · ∆t · E˙ dev,i,d,t dev = d
t
i
11
∀dev ∈ {BOI, CHP }
(6)
ACCEPTED MANUSCRIPT
In this equation, wd stands for the weighting variable for typical demand day
283
d. Since this day represents wd other days of the original input data set, the gas
284
demand at every time is weighted appropriately. Furthermore, ∆t denotes the
285
length of each time step, which has been set to one hour in this work. However,
286
if more detailed inputs are available, the temporal resolution can be adjusted
287
accordingly.
tariff: Gann dev =
XX j
290
SC
289
The annual gas consumption is distributed among the different levels of each
Gdev j,l
∀dev ∈ {BOI, CHP }
l
gas cdem · CRF · BOI = b
XX j
291
292
(7)
The demand related costs for gas consumption of boilers are computed with:
M AN U
288
RI PT
282
var,gas GBOI j,l · cj,l
(8)
l
According to the German Energy Tax Act, a tax refund of taxgas that reduces the variable costs of cvar,gas can be obtained for CHP units [14]: j,l XX gas cdem · CRF · GCHP · cvar,gas − taxgas CHP = b j,l j,l
(9)
l
D
j
Demand related electricity costs for purchases from the grid are modeled
294
similar to the costs for natural gas. In contrast to the gas tariffs, two electricity
295
tariffs may be selected from the available set of tariffs. One electricity tariff has
296
to be chosen to satisfy plug loads:
298
299
300
EP
X
tarjel∗ = 1
(10)
j∗
In Equation 10, j ∗ denotes all electricity tariffs that can be chosen for plug
AC C
297
TE
293
loads and are not special HP tariffs. If a HP unit is installed, a special HP tariff hpt, valid for electricity required
for the HP, CFWH and EH, may be selected: X
el tarhpt ≤
X
xHP i
(11)
i
hpt 301
For all types of tariffs, a tiered pricing structure is considered. These tiered
302
pricing structures for electricity tariffs are modeled in a similar manner as for 12
ACCEPTED MANUSCRIPT
gas tariffs. At most one level can be valid and if the corresponding electricity
304
tariff is not chosen, all level variables have to be 0: X
el,lvl tarj,l = tarjel
∀j
(12)
l 305
306
RI PT
303
LB The correct level is bounded by a minimum annual consumption Etarj,l and UB a maximum annual consumption Etarj,l :
∀j, l
SC
el,lvl el,lvl LB HP PL UB tarj,l · Etarj,l ≤ Elj,l + Elj,l ≤ tarj,l · Etarj,l
(13)
HP PL In this equation, Elj,l and Elj,l describe the annual electricity purchases
308
for HP and the house’s plug loads at tariff j and level l. Since the HP tariff
309
PL cannot be used to cover the house’s plug loads, Elhpt,l is set to 0.
310
The annual electricity imports for types HP and PL are: ann Eltype =
X
wd · ∆t ·
X t
d
312
levels of each tariff:
type Elj,l
317
(15)
Finally, the demand related costs for electricity are computed with:
EP
XXX type
j
type Elj,l · cvar,el j,l
(16)
l
AC C
2.2.2. Metering costs
Metering costs directly follow from the chosen tariff and the resulting level: cmet type =
316
∀type ∈ {HP, P L}
l
cdem = bel · CRF · el
315
(14)
TE
XX j
314
∀type ∈ {HP, P L}
The annual amount of electricity imports is distributed among the different
ann Eltype =
313
type,imp Pd,t
D
311
M AN U
307
XX j
type,lvl tarj,l · cfix,type j,l
∀type ∈ {el, gas}
(17)
l
In this equation, cfix,type denotes the fixed, annual metering costs of tariff j j,l
and level l.
13
ACCEPTED MANUSCRIPT
318
2.2.3. EEG levy In Germany, end-customers pay the so-called EEG levy to support RES.
320
According to the EEG [11], a certain percentage of the EEG levy has to be
321
paid to the transmission system operator for self-consumed electricity, if the
322
installed capacity exceeds 10 kW. Equations 18 and 19 describe whether or not
323
more than 10 kW capacity of CHP or PV are installed. In these equations, xeeg
324
denotes a binary variable that is equal to 1 if more than 10 kW are installed and
325
Meeg,cap is a big-M, serving as an artificial upper bound for installed generation
326
capacity, set to 1000 kW.
M AN U
X
SC
RI PT
319
nom PPV,i · zPV,i ≤ 10 + xeeg · Meeg,cap
(18)
nom PCHP,i · xCHP,i ≤ 10 + xeeg · Meeg,cap
(19)
i
X i
eeg The following two equations describe the resulting costs c˜dev if this 10 kW
threshold is not enforced: X
wd ·
X
D
eeg c˜PV = f eeg · ∆t ·
d
X
wd ·
TE
eeg c˜CHP = f eeg · ∆t ·
d
PL HP PPV,d,t + PPV,d,t
(20)
t
XX t
PL HP PCHP,i,d,t + PCHP,i,d,t
(21)
i
PL HP In these equations, Pdev and Pdev describe the portions of self-generated elec-
328
tricity that are self-consumed by the house’s plug loads and HP operation and
329
are thus not fed into the grid. Additionally, f eeg = 0.02752 Euro/kWh is the
330
cost for self-consumed electricity in 2017, which is currently equal to 40% of the
331
total EEG levy [11, 50].
AC C
EP
327
The following three equations represent the linearized formulation of the
eeg product of c˜dev · xeeg [51] and therefore describe the billed EEG levy under
consideration of the 10 kW threshold: eeg cdev ≤ xeeg · Meeg,billed
∀dev ∈ {CHP ; P V }
(22)
eeg eeg c˜dev − cdev ≥0
∀dev ∈ {CHP ; P V }
(23)
eeg eeg c˜dev − cdev ≤ (1 − xeeg ) · Meeg,billed
∀dev ∈ {CHP ; P V }
(24)
14
ACCEPTED MANUSCRIPT
In this reformulation, Meeg,billed is an upper bound for the total amount of billed
333
EEG levy, which has been set to 1,000,000.00 Euro/a.
334
2.2.4. Revenues from feed-in
RI PT
332
Furthermore, according to the EEG [11], the remuneration rate for elec-
tricity from PV that is fed into the grid depends on the installed capacity. Below 10 kW, feed-in is remunerated with pfeed,PV,10 = 0.1264 Euro/kWh, be-
SC
tween 10 and 40 kW pfeed,PV,40 = 0.123 Euro/kWh and between 40 and 750 kW
pfeed,PV,750 = 0.1103 Euro/kWh [52]. The following equations are used for de-
X
M AN U
termining the correct interval:
nom PPV,i · zPV,i ≤ 10 · xeeg,10 + 40 · xeeg,40 + 750 · xeeg,750
i
X
xPV,i = xeeg,10 + xeeg,40 + xeeg,750
i
(25) (26)
335
The total amount of sold electricity is split into the corresponding categories,
336
sell,PV where El40 for instance describes the amount of electricity sold at pfeed,PV,40 .
X
337
X t
341
342
EP
∀size ∈ {10, 40, 750}
(28)
Finally, the feed-in remuneration for PV results in:
AC C
340
(27)
Additionally, at most one of these three variables can be unequal to zero:
X
EEX efeed · CRF · PV = b
339
sell,PV Elsize
size∈{10,40,750}
sell,PV Elsize ≤ MPV,sell · xeeg,size
338
X
sell PPV,d,t =
TE
d
wd ·
D
sell,PV Eltotal = ∆t ·
sell,PV pfeed,PV,size · Elsize
(29)
size∈{10,40,750}
Electricity surplus from CHP can be fed into the grid at standard mar-
ket rates. In this work, we use a time-independent average market rate of pfeed,CHP = 0.038 Euro/kWh. This value represents the average CHP index in 2016 [53] and avoided grid costs. EEX efeed · CRF · pfeed,CHP · ∆t · CHP = b
XX i
15
d
wd ·
X t
sell PCHP,i,d,t
(30)
ACCEPTED MANUSCRIPT
344
345
346
347
2.2.5. Subsidies In this paper, we consider subsidies for CHP units according to [12] and subsidies for BAT systems based on [13].
RI PT
343
For large micro CHP units above 2 kW rated electrical power (subset i∗ ), the subsidies are computed as:
i XX X h sub,CHP sub,CHP EEX PL HP sell esub ·CRF ·∆t· wd · pself · PCHP,i · PCHP,i ∗ ,d,t + PCHP,i∗ ,d,t + p ∗ ,d,t CHP,large = b sell
348
In this equation,
d
t
psub,CHP self
SC
i∗
(31)
= 0.04 Euro/kWh denotes the subsidies for self-
consumed electricity from CHP units and psub,CHP = 0.08 Euro/kWh the subsell
350
sidies for sold power [12].
351
352
M AN U
349
Smaller micro CHP units (subset i∗ ) can either receive a fixed or a variable subsidy. The fixed subsidies are computed with: subfix = CRF · tmax · psub,CHP · fix
X
nom xCHP,i∗ · PCHP,i ∗
(32)
i∗
354
Here, tmax stands for the maximum subsidized time period of 60,000 full load hours and psub,CHP = 0.04 Euro/kWh is the specific subsidy. fix
D
353
TE
Variable subsidies for small scale CHP units are computed just like for large scale CHPs as described in Equation 31, however they are stored in variable
AC C
ner [51].
EP
subvar . Since the maximum of both, subfix and subvar will be used by investors, fix var the following equations model esub in a linear manCHP,small = max sub ; sub
fix esub CHP,small ≥ sub
(33)
var esub CHP,small ≥ sub
(34)
fix esub + M fix · δ var CHP,small ≤ sub
(35)
var esub + M var · (1 − δ var ) CHP,small ≤ sub
(36)
355
In this set of equations, M fix = CRF · tmax · psub,CHP · 2 kW is an upper fix
356
bound for subfix , M var = bEEX · CRF · 8760 h · psub,CHP · 2 kW for subvar and sell
357
δ var denotes a binary variable that is 1 if the variable subsidy option is chosen.
16
ACCEPTED MANUSCRIPT
The overall subsidies for CHP units result in:
358
sub sub esub CHP = eCHP,small + eCHP,large
RI PT
(37)
Batteries are subsidized by the German Reconstruction Credit Institute (KfW) [13]. This subsidy is essentially a relatively cheap credit for the installation of BATs in combination with PV modules. Since only 81% of this
SC
credit have to be repaid, we consider the remaining 19% of this credit as a sub-
sidy for BAT storages [13]. The subsidies are capped by the minimum of two factors. These subsidies are limited by submax BAT = 2, 000.00 Euro/kW times the
M AN U
installed PV peak power and they also do not exceed the actual investments reduced by subBAT = 1, 600.00 Euro/kW times the installed PV peak power: max sub esub BAT ≤ 19% · CRF · subBAT · ζBAT
inv inv sub esub BAT ≤ 19% · cPV + cBAT − CRF · subBAT · ζBAT
(38)
(39)
sub In this set of equations, ζBAT stands for the product of installed PV power P nom P PPV,i · zPV,i and the decision if a battery storage is installed xBAT,i . i
D
i
According to Williams [51], this product is linearized as follows: X
nom PPV,i · zPV,i
(40)
TE
sub ζBAT ≤
i
sub ζBAT
≥
EP
X Amax sub nom ζBAT ≤ max PPV,j · · xBAT,i j APV,j i X
nom PPV,i
· zPV,i −
359
360
361
362
! 1−
X
xBAT,i
i
Amax nom · max PPV,j · j APV,j
(42)
AC C
i
(41)
2.3. Technical constraints This section describes the technical constraints, which model the device se-
lection and their operation. 2.3.1. Device selection
363
The nominal heat provided by CHP, EH, boiler and HP has to match or ex-
364
ceed the design heat load (DHL) to ensure thermal comfort during cold weather
17
ACCEPTED MANUSCRIPT
conditions: X
X dev∈{BOI,CHP,EH,HP }
366
˙ DHL xdev,i · Q˙ nom dev,i ≥ Q
(43)
RI PT
365
i
2.3.2. Generating devices
Similar to Wakui and Yokoyama [41] and Wakui et al. [42], we also use a
368
piecewise linearization for modeling the part load behavior of CHP units and
369
HP units but also apply this technique to boilers. The corresponding equations
370
are presented in Appendix B. The operation of CFWHs and EHs is simplified
371
as we assume a loss-free conversion from electricity to heat and no minimum
372
activation limits during part load operation.
M AN U
SC
367
373
For the heat generation of STC modules, an efficiency ηSTC,i,d,t is computed
374
that accounts for the optical efficiency η0 as well as linear k1 and quadratic k2
375
thermal losses [54]: ηSTC,i,d,t = η0 − k1 ·
376
difference ∆TST C describes the difference between the average collector fluid
377
amb temperature and ambient temperature Td,t . In this work, we assume a constant
378
STC flow temperature of 35◦ C. The solar irradiation onto the collector is Id,t
379
which is computed according to [54, 55].
− k2 ·
2 ∆TST C Id,t .
The temperature
D
∆TST C Id,t
The electricity generation from PV is modeled similarly, considering the aver-
381
age efficiency of inverters η¯INV . Since the nominal efficiencies of all inverters con-
382
sidered in this paper vary between 97% and 98%, an average efficiency of 97.7% is
383
used for all inverters. Furthermore, the module’s efficiency is based on Dubey et
386
387
388
389
EP
385
al. [56], considering the solar irradiation onto the module as well as temperature h i Id,t amb ref amb effects: ηPV,i,d,t = η0 · 1 − γ · Td,t − TPV,i + T NOCT − Td,t · I NOCT . In
AC C
384
TE
380
this equation, γ is the loss coefficient, T NOCT and I NOCT stand for the temref perature and irradiation at NOCT conditions (20◦ C, 0.8 kW/m2 ) and TPV,i
describes the cell temperature at these conditions. The power from PV, CHP and the power discharging the BAT is split into
390
self-consumption for general plug loads (PL), the HP and electricity fed into
391
the grid (sell). This splitting is necessary for assigning the right amounts of
392
electricity to each tariff. For BAT and PV, these parts can be combined for all
18
ACCEPTED MANUSCRIPT
types of batteries and modules: X
PL HP sell Pdev,i,d,t = Pdev,d,t + Pdev,d,t + Pdev,d,t
∀dev ∈ {BAT, P V } , d, t (44)
i 394
395
RI PT
393
In case CHP units are chosen, a strict distinction between each type of CHP electricity has to be made due to the precise modeling of the KWKG: PL HP sell PCHP,i,d,t = PCHP,i,d,t + PCHP,i,d,t + PCHP,i,d,t
∀i, d, t
(45)
According to the EEG [11], at most 70% of the PV peak power may be fed
397
into the distribution grid. If a battery system is installed and the subsidies described in Section 2.2.5 are chosen, this number is even reduced to 50% [13]. h i X sell nom PPV,d,t ≤ PPV,i 0.7 · zPV,i − ζiPV,BAT + 0.5 · ζiPV,BAT ∀d, t (46)
M AN U
398
SC
396
i
In Equation 46, ζiPV,BAT stands for the product of zPV,i and is linearized as follows [51]: ζiPV,BAT ≤ zPV,i A · APV,i
X
xBAT,j
D
ζiPV,BAT ≤
max
TE
ζiPV,BAT ≥ zPV,i − 1 −
X
xBAT,j ·
j
EP
nom zPV,i · PPV,i ≤
X
i
402
403
(47)
∀i
(48)
Amax APV,i
∀i
(49)
nom xINV,j · PINV,j ·
(50)
j
2.3.3. Storages
AC C
401
∀i
The inverter is sized according to the installed, nominal PV power: X
400
xBAT,j that
j
j
399
P
Storage units’ energy contents are modeled based on their state of charge
Sdev,i,d,t . Storages can only be charged (Sdev,i,d,t > 0) if the corresponding
device has been purchased: Sdev,i,d,t ≤ xdev,i
∀dev ∈ {BAT, T ES} i, d, t
(51)
404
In order to allow a discrete selection of storage units, an energy balance
405
is modeled for each type and each time step. In previous works that used a 19
ACCEPTED MANUSCRIPT
continuous device modeling for TES units, only one equation for each time
407
step is necessary, whereas the discrete device modeling increases the amount of
408
required ‘state of charge’-variables by the number of considered TES and BAT
409
units.
Sdev,i,d,t = (1 − ϕdev,i )·Sdev,i,d,t−1 +∆t·
ηdev,i · chdev,i,d,t − dchdev,i,d,t capdev,i
RI PT
406
∀dev ∈ {BAT, T ES} i, d, t (52)
The storage’s relative standby losses between two consecutive time steps are
411
ϕdev,i , ηdev,i describes the storage cycle’s efficiency, chdev,i,d,t and dchdev,i,d,t
412
stand for the charging and discharging power and capdev,i for the storage’s
413
capacity. The capacity can directly be derived from data sheets for BATs. For
414
TES units, we calculate capdev,i = ρ·κ·Vi ·∆T max , where ρ and κ are the density
415
and specific heat capacity of water, Vi is the storage’s volume and ∆T max = 35 K
416
the maximum temperature spread inside the tank.
M AN U
SC
410
Since we are using typical demand days that are weighted with weighting
418
factors to represent a whole year, it is necessary to introduce cycling conditions.
419
init These state that the storage’s initial storage level at each day Sdev,i,d has to
420
init be equal to its final level at the end of the day. In our model, Sdev,i,d is also
421
considered as a decision variable that is optimized.
TE
D
417
init Sdev,i,d,−1 = Sdev,i,d,tend = Sdev,i,d
∀dev ∈ {BAT, T ES} i, d
(53)
Charging and discharging powers are restricted with big-M formulations,
423
preventing the charging and discharging of tanks that are not installed. Appro-
424
priate values for Mdev,i
EP
422
ch/dch
426
427
428
limits given in the data sheets, whereas multiples of Q˙ DHL are used for TES
AC C
425
for BATs are the maximum charging and discharging
units.
ch chdev,i,d,t ≤ xdev,i · Mdev,i
∀dev ∈ {BAT, T ES} , i
(54)
dch dchdev,i,d,t ≤ xdev,i · Mdev,i
∀dev ∈ {BAT, T ES} , i
(55)
For TES units, the following balances are applied to determine the charging and discharging power: 20
ACCEPTED MANUSCRIPT
chTES,i,d,t =
dev ∗
i
X
XX
Q˙ dev∗ ,j,d,t
∀d, t
j
dchTES,i,d,t = Q˙ DHW + Q˙ SH d,t d,t
∀d, t
i
(57)
In these formulations, dev ∗ describes the subsets of all heat generating de-
430
vices, Q˙ DHW and Q˙ SH d,t d,t describe the building’s domestic hot water and space
431
heating demands at time t on day d. Both equations, in combination with the
432
big-M constraints, ensure that all generated and consumed heat are interchanged
433
with exactly one thermal storage tank.
M AN U
SC
429
434
Since we assume DHW to be heated up from 10◦ C to 60◦ C and both STC
435
and HP to only provide heat at 35◦ C, the remaining temperature lift has to be
436
provided by other heat generators than HP or STC. Due to the usage of typical
437
demand days, these constraints are formulated for the entire day and not for
X
each hour, allowing for a flexible charging and discharging strategy: X X X 60 − 35 X ˙ Q˙ dev,i,d,t ≥ · QDHW,d,t · xdev,i ∀dev ∈ {HP, ST C} , d 60 − 10 t i i
D
438
(58)
TE
dev∈{BOI,CF W H,CHP,EH}
Additionally, CFWHs are assumed to have a small storage volume in com-
440
parison with the TES unit, therefore the following equation limits the heat
441
output of the CFWH:
EP
439
X
Q˙ CF W H,i,d,t ≤ Q˙ DHW,d,t
∀i, d, t
(59)
i
442
443
444
445
AC C
t
(56)
RI PT
X
For BAT systems, the charging and discharging powers follow from the build-
ing’s electricity balances. We formulate one balance for the building’s plug loads and a second balance for the electricity potentially billed with a special heat pump tariff. The building’s electricity balance is written as: X X P L,imp PL PL PL PL PL PL Pd,t +PCFWH,d,t +PEH,d,t + chBAT,i,d,t = Pd,t +PPV,d,t +PBAT,d,t + PCHP,i,d,t i
i
(60)
21
∀d, t
ACCEPTED MANUSCRIPT
PL PL PL Here, Pd,t describes the house’s plug loads, PCFWH,d,t and PEH,d,t stand for
447
the amount of electricity consumed by an CFWH or EH that is not billed under
448
a special HP tariff.
RI PT
446
The second electricity balance is:
449
X
X HP,imp HP HP HP HP HP PHP,i,d,t +PCFWH,d,t +PEH,d,t = Pd,t +PPV,d,t +PBAT,d,t + PCHP,i,d,t
i
i
(61)
The CFWH’s and EH’s electricity amounts are further defined: In total,
451
the amount of electricity purchased at a potential HP tariff and the amount
452
purchased at the standard tariff, have to be equal to the electricity consumption caused by this device: X PL HP Pdev,i,d,t = Pdev,d,t + Pdev,d,t
M AN U
453
i 454
455
SC
450
∀dev ∈ {CF W H, EH} , d, t
(62)
If no HP has been purchased, the CFWH and EH cannot be active in the second electricity balance: nom X nom HP HP · xHP,i PCFWH,d,t + PEH,d,t ≤ max PCFWH,j + max PEH,j j
j
∀d, t
i
D
(63) Finally, we prevent the HP and STC from being activated if the thermal
TE
storage’s average temperature is above the set flow temperature of 35◦ C. We assume that both can provide a temperature spread of 10 K and that the storage
456
457
458
AC C
EP
is ideally stratified. X ∆T HP STES,i,d,t ≤ 1 − yHP,j,d,t · 1 − ∆T max i X ∆T STC STES,i,d,t ≤ 1 − ySTC,j,d,t · 1 − ∆T max i
∀j, d, t
(64)
∀j, d, t
(65)
2.4. Ecological constraints In line with the Paris Agreement [57], we use CO2 emissions as an index for
measuring the ecologic impact of the optimal energy systems. Annual CO2 emis-
459
sions comprise emissions from gas usage emigas , imported electricity emiel,imp
460
and negative emissions from electricity exports emiel,exp . emiann = emigas + emiel,imp − emiel,exp 22
(66)
∀d, t
ACCEPTED MANUSCRIPT
are assumed to be time-independent, the emissions from natural gas result in: emigas =
X
emigas,spec · j
X
j 463
l
X
emiel,spec · j
X
j
PL HP Elj,l + Elj,l
l
CO2 [58]: = 0.527 kgkWh tricity mix which causes emiel,spec EM
"
el,exp
emi
=
emiel,spec EM
·
X
wd · ∆t ·
d
466
(67)
(68)
Electricity fed into the public grid is expected to replace the average elec-
M AN U
465
Similarly, CO2 emissions from electricity imports are modeled: emiel,imp =
464
CHP GBOI j,l + Gj,l
RI PT
462
Considering the specific CO2 emissions emigas,spec for each tariff j, which j
SC
461
2.5. Solution algorithm
!
X
X
t
i
sell PCHP,i,d,t
+
# sell PPV,d,t
+
sell PBAT,d,t
(69)
The described model defines a mixed integer linear program that can be
468
solved with existing solvers. In this work, all computations are carried out with
469
Gurobi 7.02 and all models are set up with the corresponding Gurobi-Python
470
framework (gurobipy). We used the standard Branch-and-Cut [59] algorithms
471
from this solver.
TE
D
467
Since the calculating times of the described MILP problem easily become
473
intractable with increasing typical demand days, time resolution, and available
474
devices, we implemented a decomposition approach that has also been proposed
475
by Wakui and Yokoyama [60]. In this approach, the solution space is reduced by
477
478
479
480
AC C
476
EP
472
prescribing the device selection. Additionally, the objective function is forced to be lower than the previously determined minimum objective value. This limit on the objective additionally accelerates the solution process, since device candidates that cannot improve the objective are quickly marked as infeasible and therefore solving until optimality is omitted. 2 http://www.gurobi.com/index
23
ACCEPTED MANUSCRIPT
In our implementation of this approach, each major conventional heat gen-
482
erator (gas boiler, CHP unit and HP unit) is consecutively fixed, starting with
483
the smallest gas boiler and finishing with the largest HP unit.
484
3. Setup
RI PT
481
This section summarizes the used inputs and describes the conducted calcu-
486
lations. Further information on these inputs as well as the implemented model
487
can be found in this project’s open-source repository3 .
488
3.1. Inputs
M AN U
SC
485
489
The model’s input time series consist of solar irradiation onto the roof area
490
and the building’s energy load profiles, such as electricity demands of plug loads,
491
domestic hot water usage, and space heating loads. Additionally, the considered
492
energy conversion units are described in this subsection.
493
3.1.1. Energy load profiles
We apply the optimization model described in the previous section to three
495
newly constructed residential buildings significantly varying in size that are
496
located in different regions of Germany. The first building is a small single
497
family house located in Bavaria, the second building is a medium-sized multi
498
family house in Hamburg, and the third building represents a large apartment
499
building in Berlin. The key parameters are summarized in Table C.5. For all
500
roofs solar modules are assumed to be oriented southwards and elevated at a
501
pitch angle of 35◦ .
503
504
TE
EP
AC C
502
D
494
Time series for ambient temperature and solar irradiation for calculating
the building’s heat demand and PV as well as STC efficiencies and outputs are taken from German Test Reference Years [61] for the corresponding regions. 3 https://github.com/RWTH-EBC/BESopt
24
ACCEPTED MANUSCRIPT
The hourly space heating profiles are computed with the freely available
506
building simulation library AixLib [62]4 and the building models are param-
507
eterized with the also open-source available software package TEASER [63]5 .
508
Electricity demands for non-heating devices, appliances and lighting are com-
509
puted with a high-resolution, stochastic tool based on Richardson et al. [64].
510
Domestic hot water demand profiles are calculated with a combination of the
511
users’ occupancy based on Richardson et al. [64] and daily tap water usage
512
statistics of residential buildings developed in IEA Annex 42 [65]. The cumu-
513
lated, annual domestic hot water, space heating and electricity demands are also
514
listed in Table C.5.
M AN U
SC
RI PT
505
As mentioned previously, using full year inputs leads to computationally
516
intractable simulations. Therefore, the input time series are reduced to 5 typical
517
demand days with hourly time resolution, by using k-medoids clustering [43, 44,
518
45]. We have chosen 5 typical demand days since the original building energy
519
system optimization model also used 5 days, however these demand days were
520
determined differently [41, 42].
521
3.1.2. Energy conversion units
D
515
All energy conversion units that have been modeled are based on manu-
523
facturers’ data sheets. Price recommendations are taken from multiple online
524
retailers or original manufacturers for all devices but CHP units. For CHP units,
525
the regression curves provided by ASUE [49] have been used for estimating CHP
526
units’ investment costs.
528
529
530
531
EP
Table C.6 displays the characteristics of all used gas boilers and CHP units.
AC C
527
TE
522
In this table, all used node points for the interpolation of heat and power output as well as gas consumption during part load operation are listed. We assume that each device can operate flexibly between these given points. The index ‘mds’ refers to manufacturer’s data sheets. 4 https://github.com/RWTH-EBC/AixLib 5 https://github.com/RWTH-EBC/TEASER
25
ACCEPTED MANUSCRIPT
Table C.7 shows the considered CFWHs, EHs and HPs. The nominal heat
533
output Q˙ nom and electricity input P nom of HPs describe the device’s operation
534
at approx. 7 ◦ C outdoor temperature and 35 ◦ C flow temperature.
RI PT
532
The characteristics displayed in Table C.8 have been used for all HPs. The
536
left column represents the ambient temperature and the right columns the device
537
operation, starting with the full load operation. In each cell, the left entry
538
represents the scaled heat output and the right entry the also scaled electricity
539
consumption. The scaling factors are equal to the nominal heat outputs shown
540
in Table C.7. We assume that HPs are primarily used for covering space heating
541
and therefore provide heat at 35◦ C at all times.
543
M AN U
542
SC
535
The considered STCs are listed in Table C.9, PV modules and INVs are shown in Table C.10.
Table C.11 displays the available BAT storages and TES units. BAT’s ca-
545
pacity is given in kWh and for TES, m3 is used. Furthermore, BAT’s capacity
546
describes the effective capacity, including the maximum allowed depth of dis-
547
charge. TES units’ self-discharge is considered through the loss coefficient ϕ
548
that describes the energy loss during 24 hours of standby operation. In con-
549
trast, self-discharge is neglected for BATs [66].
550
3.1.3. Tariffs and emissions
TE
D
544
The electricity and gas tariffs’ characteristics are listed in Tables C.12. All
552
tariffs are derived from a local utility provider [67]. The specific emissions for
553
standard and HP tariffs are based on [58]. Price information and the tiered
554
pricing structure are taken from [67]. Regarding green tariffs, we assumed a
556
557
558
AC C
555
EP
551
share of 45% of renewable electricity generation and the remaining share to be generated conventionally but covered with internationally available renewable certificates [68, 69]. These remaining 55% are therefore assumed to cause local emissions that are equal to the average electricity mix. For eco gas, a 10% share
559
of biogas is assumed to increase the variable costs by 0.0060 Euro/kWh [70]. As
560
shown in both tables, standard tariffs are slightly less expensive than eco-tariffs,
561
but cause significantly more CO2 emissions. The implemented HP tariff further
26
ACCEPTED MANUSCRIPT
offers a strong economic incentive compared to the other tariffs.
563
3.2. Economic parameters
RI PT
562
Table C.13 lists the economic parameters used in this study. The price
565
change factors for electricity and gas are derived by linear regressions from the
566
average German prices between 2008 and 2016 [71]. Similarly, the price change
567
factor for EEX compensation is derived by linear regression of the CHP indexes
568
of the past ten years [53]. The expected lifetimes of the devices are based on
569
VDI 2067 [46].
570
3.3. Calculations
M AN U
SC
564
571
The calculations conducted in this work aim at comparing the influence of
572
the discrete modeling of storage units, assessing the importance of modeling
573
subsidies and market characteristics and analyzing the robustness of our model.
574
3.3.1. Discrete storage modeling
In order to evaluate the importance of a discrete storage selection, we con-
576
duct a total of six optimizations for each building. These consist of a cost
577
minimization for the new and original model as well as a recalculation. Since
578
the storage units are expected to deviate due to the different modeling, the re-
579
calculation of the original model is conducted, enforcing the optimal solution of
580
the new model. Additionally, a second set of three optimizations is conducted
581
with an enforced reduction of CO2 emissions by 20%.
582
3.3.2. Importance of subsidy modeling
584
585
586
TE
EP
AC C
583
D
575
For assessing the importance of accounting for subsidies, we compare the
results of our model for the cost minimization and enforced CO2 reduction with
a model neglecting all previously mentioned subsidies and regulations. In this case, ceeg and esub are set to zero, and the feed-in limit of 70% of the installed
587
peak power for PV modules is omitted. Additionally, we assume that electricity
588
from PV units is sold at the same market price as electricity from CHP units
589
and that only standard tariffs are available. 27
ACCEPTED MANUSCRIPT
590
Thus, we analyze and compare three different models. The total amount of constraints and variables for these models are summarized in Table C.14.
592
3.3.3. Sensitivity analysis
RI PT
591
The model relies on a number of uncertain factors, for instance weather
594
conditions, user behavior, energy tariffs, device operation and investment costs.
595
Weather conditions and user behavior both strongly affect the energy load
596
profiles, such as household electricity, domestic hot water and space heating
597
demands. These effects can be analyzed and taken into consideration through
598
various different methods, such as weather data from different years, extreme
599
weather scenarios, and complex user behavior models. In this work, we account
600
for these effects, in a simplified manner by linearly scaling the original demands
601
with 0.95 (-5% demands scenario) and 1.05 (+5% demands scenario).
M AN U
SC
593
Additionally, the chosen tariffs are expected to significantly impact the opti-
603
mization results. Therefore, we also analyze a reduced and a high tariff scenario,
604
by linearly scaling the original electricity and gas tariffs’ variable costs with 0.95
605
(-5% tariffs scenario) and 1.05 (+5% tariffs scenario).
D
602
Device operation is uncertain since it depends on the quality of the instal-
607
lation, for instance the system’s hydraulic balance. Investment and installation
608
costs are also considered to be uncertain, since they strongly vary regionally and
609
they are typically highly influenced by the customer’s market power and nego-
610
tiation abilities. Since we assume that device operation is of less importance
611
than the other influences and that investment costs can be determined precisely
612
when applying the model for real life applications based on price inquiries, we
614
615
616
617
EP
AC C
613
TE
606
do not include these factors in our sensitivity analysis. Similar to the influence analysis of the discrete storage model, we also con-
duct a new optimization run for each scenario of the uncertainty analysis as well as a recalculation that enforces the base case’s results given the changed demands or tariffs.
28
ACCEPTED MANUSCRIPT
618
3.4. Computing hardware R R For all optimizations, we used a Windows 7 computer with an Intel Xeon
620
E5-2630 v2 CPU utilizing 6 threads and 32 GB of RAM. All optimizations are
621
solved to an optimality gap of 1% and all recalculations are solved with a gap
622
of 0.1%. Since most inputs are uncertain and the model also still presents
623
simplifications that will be addressed in Section 4.5, an optimality gap of 1% is
624
assumed to be reasonable.
625
4. Results
SC
RI PT
619
First, the discrete storage tank modeling and the importance of accounting
627
for subsidies are evaluated for all three buildings. Subsequently, the sensitivity
628
analysis is presented and the limitations of our model are discussed.
629
4.1. Single family house
M AN U
626
The key results of cost minimizations and forced CO2 reductions for the
631
single family house are summarized in Table C.15. The left part shows the cost
632
minimization results whereas the right part lists the key findings of the forced
633
CO2 reductions. The columns labeled ‘new model’ stand for our developed
634
model, whereas ‘original’ marks the original model based on Wakui et al. [42] and
635
’no subsidies’ represents our modeling approach without considering subsidies
636
and regulations. The original formulation calculations are further split into a
637
real optimization (opt.) of the energy system design and its operation as well
638
as a recalculation (rec.). In the recalculation, the optimal energy system of
639
the new model is imposed on this model, leaving the system’s operation as the
641
642
643
644
TE
EP
AC C
640
D
630
remaining decision variables. In all cases, neither BATs, CFWHs, CHPs nor HPs have been chosen. Ad-
ditionally, the installed PV capacities are below 10 kW, therefore, no EEG levy has to be paid. Furthermore, all calculations lead to standard gas and electricity tariffs.
645
The influence of discrete storage modeling can be seen by comparing the
646
new model with the original formulation. For the single family house, the cost 29
ACCEPTED MANUSCRIPT
optimal energy system consists of a small boiler that is supported by one STC
648
module of type 3. The remaining roof area is covered with 37.62 m2 of PV
649
units, which is equivalent to 23 modules of type 3. In the new model with
650
discrete storage tank modeling, a tank with 0.49 m3 volume is selected, whereas
651
the continuous storage modeling in the original model leads to a smaller TES
652
with 0.33 m3 . Consequently, the original model requires lower investment costs
653
but leads to higher demand related costs since less heat from the STC can be
654
utilized. The recalculation shows that the modeling of TES units’ investment
655
costs causes a deviation of 1.88 Euro/a. In total, the costs of the new and original
656
model differ by less than 0.5%. Due to the large PV area and the resulting high
657
amount of PV feed-in, negative CO2 emissions occur that consequently lead to
658
the same energy systems when enforcing a relative reduction of CO2 emissions
659
by 20%.
M AN U
SC
RI PT
647
In contrast, without subsidies an additional EH is installed and the PV
661
area is strongly reduced. As a consequence, less PV is exported leading to lower
662
negative CO2 emissions. When enforcing the CO2 reduction, TES size as well as
663
the amount of PV modules and STCs are increased. The larger TES is necessary
664
for integrating more heat from the STCs. Since more heat is generated through
665
solar energy, less gas is required from the boiler which switches the gas tariff from
666
level 2 to level 1, strongly reducing the fixed metering costs. Table C.15 also
667
shows that without subsidies, the revenues from electricity feed-in are strongly
668
reduced, since less PV modules are installed and feed-in is only remunerated with
669
the average market rate and not the high remuneration according to the EEG.
670
The comparison with the original model shows that accounting for available
672
673
674
675
TE
EP
AC C
671
D
660
subsidies and market characteristics leads to cost benefits of 278.26 Euro/a (14.8%) and 285.46 Euro/a (15.2%) for the forced CO2 reductions. For the single family house, the new model requires approx. 660 seconds of
calculating time, the original model 1340 seconds and the new model without subsidies needs 750 and 850 seconds.
30
ACCEPTED MANUSCRIPT
676
4.2. Multi family house Similar to the single family house, Table C.16 displays the results for the
678
multi family building. The original and new model formulations again lead to
679
very similar results and only slightly differ in the TES sizes for both, the cost
680
minimization and the CO2 reduction. The optimal energy systems consist of a
681
medium sized CHP unit with 12.5 kW heat output in combination with an EH
682
that functions as a backup unit and is necessary for meeting the design heat load.
683
Additionally, a TES unit with 0.89-0.98 m3 and a large PV area are installed. In
684
the cost minimization case, the PV area has a peak output of 9.99 kW, therefore
685
no EEG levy has to be paid. In contrast, when enforcing the CO2 reduction,
686
the PV area is enlarged, leading to EEG levies of approx. 460 Euro/a (roughly
687
11% of the total annual costs). The comparison of the storage modules shows
688
that the new model predicts slightly larger storage capacities. The modeling of
689
the TES investment costs leads to deviations of less than 3.77 Euro/a and total
690
costs also only deviate by less than 15.45 Euro/a (0.21%).
M AN U
SC
RI PT
677
The no subsidies cases lead to energy systems that strongly utilize gas boil-
692
ers for this building. The cost minimal solution uses a 20.3 kW gas boiler in
693
combination with an 8 kW EH and a 0.75 m3 TES unit. Additionally, a 22 kWh
694
BAT is installed and the entire available roof area is covered with PV modules.
695
When enforcing CO2 reductions, a small CHP unit is installed and 4 modules of
696
STC type 3 replace a portion of the PV area. The cost optimal solution strongly
697
focuses on self-consumption, therefore the BAT and EH are installed to capital-
698
ize on PV generated electricity. By using electricity for heating purposes, heat
699
can be produced at 0.0380 Euro/kWh (missed remuneration for avoided grid
701
702
703
TE
EP
AC C
700
D
691
feed-in), instead of more than 0.0615 Euro/kWh when using the gas boiler. Unlike for the single family house, the consideration of subsidies increases
the carbon footprint in the cost minimization. More PV modules are built and less grid electricity is needed when neglecting subsidies in this case, leading
704
to lower total emissions. This is obviously in contradiction with the intention
705
of the modeled subsidies. However, when enforcing CO2 reductions, more PV
706
units are used in the new and original model, which in turn leads to lower 31
ACCEPTED MANUSCRIPT
CO2 emissions than in the no subsidies calculation. Additionally, the solutions
708
of the CO2 reduced scenario are cheaper and generate less CO2 than the cost
709
minimum without these subsidies. This illustrates that subsidies still support
710
CO2 reductions and provide benefits for the energy system’s owner.
RI PT
707
The calculating times for the multi family house are approx. 2935 and 1580
712
seconds with the new model, 1970 and 710 seconds with the original formulation
713
and 1450 as well as 1780 seconds without considering subsidies and market
714
characteristics.
715
4.3. Apartment building
M AN U
SC
711
Table C.17 displays the results for the apartment building. The cost optimal
717
energy systems obtained when considering subsidies and market characteristics,
718
comprise a medium sized CHP unit that is supported by a boiler and an EH
719
to meet the design heat load. The TES sizes vary between 1.50 m3 in the new
720
model and 1.36 m3 in the original model formulation. In all cases, the available
721
roof area is covered with PV modules exclusively, exceeding the 10 kW threshold
722
which leads to EEG levy charges between approx. 750 and 850 Euro/a. When
723
requiring CO2 reductions, a HP unit replaces the EH. Additionally, the gas
724
boiler is slightly enlarged for covering the design heat load. In contrast to the
725
other simulations, the electricity tariff is also used as a measure for reducing
726
CO2 emissions. When reducing CO2 emissions, an eco tariff is used, whereas
727
in the cost optimal solutions a standard tariff is employed. In the reduced CO2
728
case, the sizes of the CHP units and consequently also of the TES units are
729
enlarged. The original model formulation requires a TES with 1.85 m3 volume,
731
732
733
TE
EP
AC C
730
D
716
whereas the new formulation uses a 2.00 m3 tank. Additionally, 1 STC module
is installed in the new model and 2 modules are used in the optimal energy system based on the original formulation. The different tank sizes lead to errors in the estimation of investment costs of approx. 15 Euro/a, which is considered
734
negligible in comparison with the total investment costs of approx. 6300 Euro/a.
735
Furthermore, the inaccuracy resulting from scaling a representative TES unit is
736
also not significant since the models differ by at most 0.1%. 32
ACCEPTED MANUSCRIPT
Without considering subsidies, a smaller CHP unit is required. For compen-
738
sation, a larger boiler is installed. As a consequence, the TES volume can be
739
reduced to only 0.98 m3 . Like for the multi family building, the case without
740
subsidies leads to the installation of a BAT. For this building, approx. 128 m2
741
of the roof area are covered with PV units and additionally, one STC module
742
is installed. When forcing CO2 reductions, the PV area is maxed out and more
743
PV is fed into the grid. The installation of additional 39 m2 of PV modules
744
(24 modules) only increases the total costs by 60 Euro/a, indicating that PV
745
modules are priced competitively under the used inputs and assumptions. In
746
both calculations, the overall CO2 emissions are significantly higher than in the
747
cases when accounting for subsidies. Therefore, the governmental subsidies and
748
market characteristics lead to the intended goal of significantly reducing CO2
749
emissions for this building.
M AN U
SC
RI PT
737
For the apartment building, the new model requires approx. 3285 and 4615
751
seconds, the original model 3135 and 4840 seconds, and the new model without
752
subsidies 530 and 1275 seconds. Overall, the discrete modeling of storage devices
753
does not appear to strongly influence the calculating times. When comparing
754
the calculating times for all presented optimization runs by using the geometric
755
mean, the new model improves the run times by 5%6 . However, considering
756
subsidies and market characteristics strongly increases the computing times.
757
In our analysis, the run times are increased by 71% when using the geometric
758
mean.
EP
TE
D
750
These results are counterintuitive considering the number of constraints and
760
variables in each model. All three models have roughly the same amount of
761
762
763
AC C
759
binary variables. The original model has the lowest number of constraints and total variables, whereas the new formulation requires the highest amount of both. In combination, this often suggests that the original model would have 6 The
geometric mean is used since the calculating times are on different ranges. When
comparing both with a different metric, average relative deviations, the original model has 11% lower run times.
33
ACCEPTED MANUSCRIPT
the fastest solving times. However, we believe that the solution times can be ex-
765
plained best by considering the structure of the constraint matrix. Accounting
766
for a discrete storage model hardly affects the structure of the constraint ma-
767
trix, therefore the calculating times are similar in the new and original model.
768
However, the introduction of subsidies and market characteristics adds dense
769
constraints to the model, such as Equations 20, 21, 27 and 31. According to
770
the Gurobi website [72], Gurobi and other commercial MILP solvers are able to
771
strongly exploit sparse constraint matrices, leading to low calculating times.
772
4.4. Sensitivity analysis
M AN U
SC
RI PT
764
The results of the sensitivity analysis are shown in Table C.18. For the single
774
family house, only a demand reduction leads to an adjusted cost-optimal energy
775
system. In this case, the storage unit can be sized slightly smaller. Furthermore,
776
in this case it is necessary to allow for installing a small EH, because due to the
777
reduced demands and hourly time discretization, the minimum gas boiler heat
778
output would already provide too much heat that would not allow for fulfilling
779
the storage’s cycling conditions (Equation 53). The resulting cost difference for
780
the -5% demands scenario is approx. 8.30 Euro/a which is equivalent to 0.5%.
781
In the other cases related to the SFH, the base case’s optimal energy system is
782
always a robust solution that is not altered.
TE
D
773
The resulting cost optimal energy systems for the multi family house differ
784
for the -5% demands and -5% tariffs scenarios. With lower demands, the CHP
785
unit becomes less attractive due to reduced operating times and is therefore re-
786
placed by a gas boiler. Additionally, the TES volume is decreased and two STC
788
789
790
AC C
787
EP
783
modules with a total of 4.62 m2 as well as a larger PV area are installed. In this manner, the no longer available electricity generation from CHP is partly compensated with PV modules, which reduces electricity purchases from the grid. This case poses the largest deviation between the resulting total annual costs of
791
the optimization and recalculation, amounting to 103.48 Euro/a, approx. 1.5%.
792
A second deviation occurs for the -5% tariffs case. In this setting, a similar
793
optimal energy system is found as in the scenario with reduced demands. The 34
ACCEPTED MANUSCRIPT
initially installed CHP unit becomes less economic since savings from generating
795
electricity in comparison to imports from the grid are reduced. The resulting
796
cost deviation is 0.7%.
RI PT
794
The sensitivity analysis reveals that a 5% increase in demands makes a
798
larger CHP unit preferable for the apartment building. In this case, the TES
799
unit is increased to 2.00 m3 in order to provide more operating flexibility and
800
increase the operating time of the prime mover. The costs of optimization and
801
recalculation differ by 0.5% in this setting.
SC
797
Overall, the results of the base case appear to be relatively robust. The
803
optimal energy systems are only changed in 4 out of the investigated 12 cases.
804
The prime mover is increased in capacity once and completely replaced in two of
805
these cases. The maximum cost deviation between recalculation and optimiza-
806
tion is 1.5% in the multi family case with 5% demand reduction. On average,
807
the costs between optimization and recalculation differ by 0.3%.
M AN U
802
Interestingly, despite the fact that the optimization run is always conducted
809
with a tolerated optimality gap of 1% and the recalculation has a gap of 0.1%,
810
the solutions only marginally differ by at most 0.04% if the same energy systems
811
are chosen. This suggests that the solver converges to an (almost) optimal
812
solution even at a higher tolerated gap.
813
4.5. Limitations and future work
816
817
818
819
820
821
TE
EP
815
This section critically assesses the limitations of the developed model and presents possible improvements. This paper focuses on the optimal design, sizing and operation of building
AC C
814
D
808
energy systems for residential buildings. The developed model has been applied to newly constructed buildings. The model could be applied to existing buildings with already installed devices, however this would require multiple runs in which either the heat distribution system (radiators, floor heating) would be used after refurbishment or be replaced, too.
822
Since the model is currently only intended for single buildings, regional ef-
823
fects are neglected. Considering the obtained results and assuming a wide ap35
ACCEPTED MANUSCRIPT
plication of our findings, large areas of PV units would provide high amounts
825
of electricity during sunny periods and feed them into the grid. This could
826
potentially be harmful for existing distribution grids, leading to congestion and
827
voltage fluctuations in case of abruptly changing weather conditions, e.g. caused
828
by clouds. Considering multiple buildings simultaneously could potentially lead
829
to smarter subsystems consisting of local generation and consumption units as
830
well as electricity storages.
RI PT
824
Currently, the model has been used with hourly inputs. Due to the already
832
relatively long computing times and the low availability of weather data with
833
higher resolution, hourly inputs have been used as a trade-off solution. However,
834
using hourly averaged time series strongly benefits solar generation units, since
835
fluctuations in the solar irradiation as well as fluctuating local demands are
836
balanced, leading to higher rates of self-consumption. Since the share of self-
837
consumed electricity tends to be overestimated, grid imports and consequently
838
overall costs are expected to be underestimated.
M AN U
SC
831
The presented model does not account for installation and assembly costs in
840
detail. These costs could be accounted for in a higher level of detail, e.g. module
841
specific installation costs as well as module unspecific costs, such as call out
842
charges. Including these costs would likely affect the results, in particular the
843
installation of one or two STCs would become less attractive. These costs can
844
hardly be estimated from an academic point of view since they strongly depend
845
on the building’s location, installation service, etc. However, such costs could
846
be specified precisely by installation companies using tools like our developed
847
model.
849
850
851
TE
EP
AC C
848
D
839
In addition, there are large uncertainties regarding the modeling of specific
tariff parameters and CO2 emissions. The pricing structure has largely been based on our local utility service provider, therefore tariffs could strongly deviate in other regions. Furthermore, we modeled the eco tariff for electricity assuming
852
that 45% of the total electricity is generated with RES. When choosing this
853
share of renewables, we assumed that the remaining part would be generated
854
conventionally and covered with renewable certificates [68, 69]. This assumption 36
ACCEPTED MANUSCRIPT
might not be valid for all tariffs. Therefore, future studies could investigate the
856
influence of tariffs that exclusively offer locally generated electricity from RES.
857
Moreover, we used constant emission factors for gas and electricity in this study,
858
since such data is typically available and represents current balancing standards.
859
However, accurately assessing the emissions caused by the energy system and
860
its operation, requires dynamic emission factors that reflect the entire electricity
861
generation within the market.
RI PT
855
An obvious weakness of our model is the assumption of perfect knowledge of
863
upcoming demands. We assume electricity and thermal demands as well as solar
864
irradiation to be known precisely for each considered time step. Additionally, we
865
assume these inputs to be constant for the entire observation period. In reality,
866
these demands significantly change from year to year, since the building’s usage
867
and ambient conditions change. Furthermore, even short-term load forecasts
868
suffer from inaccuracies, leading to suboptimal device scheduling. This problem
869
could be overcome by using a multi-level optimization approach [73]. On a top
870
level, the device selection could be handled, whereas a rolling horizon scheme
871
with either robust or stochastic optimization, can be used on a lower level to
872
optimize the operation.
TE
D
M AN U
SC
862
Related to the previous limitation, a longer observation period would present
874
an interesting extension for this model. In this manner, modified building usage,
875
e.g. due to changing family structures, can be taken into account and roadmaps
876
for the installation and replacement of energy system components could be
877
developed.
879
880
881
882
Furthermore, the model could be extended in future works by accounting
AC C
878
EP
873
for start, stop and ramping constraints. Currently, there are neither minimum activation nor minimum turn-off times implemented. Future works could integrate such minimum time durations in which the devices has to stay activated or deactivated in order to reduce wear and tear. Additionally, such device wear
883
and tear is currently not considered. Since the observation period already covers
884
multiple years, accounting for reduced efficiencies due to these effects is likely
885
to affect the results. 37
ACCEPTED MANUSCRIPT
The model is intended for optimizing the building’s energy system. How-
887
ever, accounting for investments into passive components such as the building’s
888
envelope is expected to also affect the energy system. Future works could there-
889
fore integrate design decisions regarding envelope components into the current
890
model.
891
5. Conclusions
SC
RI PT
886
In this paper, a MILP for the optimal structural design, sizing and operation
893
of building energy systems has been developed. The model enhances existing
894
formulations by considering specific German regulations and market character-
895
istics, and by accounting for multiple gas and electricity tariffs. Additionally,
896
we extend previous approaches by a discrete sizing of storage units.
M AN U
892
The model has been applied to three newly constructed residential buildings
898
in different locations in Germany. Accounting for a discrete sizing of storage
899
units has marginally impacted the optimal energy systems and estimated total
900
costs. However, due to the device-specific modeling, the optimal energy system
901
is ensured to be available for purchase in reality, whereas the original formulation
902
can lead to intermediate sizes that cannot be purchased (e.g. a 1.36 m3 TES
903
instead of a 0.98 m3 or 1.50 m3 device which are available). Despite the fact that
904
the proposed formulation increases the amount of constraints and variables, the
905
effect on computing times is negligible. Depending on the used metric, either
906
the new model or the original formulation is beneficial regarding run times.
908
909
910
911
912
TE
EP
On the other hand, accounting for subsidies and market characteristics has
AC C
907
D
897
shown to strongly influence the optimal energy systems. When neglecting these constraints, less PV modules but more STCs are installed. Additionally, BATs are purchased for using electricity from PV locally, reducing grid dependence in contrast to feeding PV into the grid for a low remuneration. Furthermore, CHP units become less attractive when not considering subsidies.
913
The conducted sensitivity showed that the computed optimal energy systems
914
are relatively robust regarding demand and tariff changes. Although the tariffs
38
ACCEPTED MANUSCRIPT
and demands have been varied by 5%, the total costs only deviate by 1.5% when
916
comparing the original and the new optimal solution. Both optimal solutions
917
differ in 4 out of the 12 investigated cases and in only two of these cases, the
918
energy system is changed substantially.
RI PT
915
Several limitations of our model have been identified and discussed. The
920
most important weaknesses are the focus on individual buildings, the assump-
921
tion of perfect predictions as well as the low level of detail when modeling instal-
922
lation costs. In this regard, future works could extend the model to small city
923
districts, include a multi-level optimization for dealing with uncertain demand
924
predictions and accounting for different types of installation costs. Furthermore,
925
the model can be improved by considering longer observation periods, modeling
926
start-up, shut-down and ramping constraints as well as devices’ wear and tear.
927
Additionally, a more holistic approach can be implemented in future works that
928
accounts for also optimizing passive components such as the building’s enve-
929
lope.
930
Acknowledgments
M AN U
D
TE
Initiative “Energy System 2050 - A Contribution of the Research Field Energy”.
EP
932
This work was supported by the Helmholtz Association under the Joint
AC C
931
SC
919
39
933
Appendix A. Energy balances
934
Appendix B. Additional equations
RI PT
ACCEPTED MANUSCRIPT
Modeling of investment costs: cinv dev = CRF ·
X xdev,i · (1 − rvdev,i ) · cinv dev,i
∀ dev ∈ devs\ {P V, ST C}
i
cinv dev
SC
(B.1)
X = CRF · zdev,i · (1 − rvdev,i ) · cinv dev,i
∀ dev ∈ {P V, ST C}
cinv =
X
M AN U
i
cinv dev
dev 935
(B.2) (B.3)
At most one type of each device may be chosen: X
xdev,i ≤ 1
i
∀dev
(B.4)
The available roof area cannot be exceeded:
dev
∀dev ∈ {P V, ST C} , i
zdev,i · Adev,i ≤ Amax
i
(B.5) (B.6)
TE
XX
Amax Adev,i
D
zdev,i ≤ xdev,i ·
EP
Modeling of operation and maintenance costs: infl co&m · CRF · dev = b
X
o&m xdev,i · cinv dev,i · fdev,i
∀ dev ∈ devs\ {CHP, P V, ST C}
AC C
i
infl co&m · CRF · dev = b
infl co&m · CRF · CHP = b
936
X
(B.7) o&m
zdev,i · cinv dev,i · fdev,i
∀ dev ∈ {P V, ST C}
i
(B.8) XXX i
d
o&m PCHP,i,d,t · ∆t · fCHP,i
(B.9)
t
Heating devices can only be switched on, if they have been purchased: ydev,i,d,t ≤ xdev,i
∀dev ∈ {BOI, CF W H, CHP, EH, HP } , i, d, t 40
(B.10)
ACCEPTED MANUSCRIPT
Piecewise linearization of the performance charts of boilers, CHP units and
Q˙ dev,i,d,t =
X
wdev,i,d,t,k · Q˙ mds dev,i,d,t,k
RI PT
HP units: ∀dev ∈ {BOI, CHP, HP } , i, d, t
k
(B.11)
E˙ dev,i,d,t =
X
mds wdev,i,d,t,k · E˙ dev,i,d,t,k
∀dev ∈ {BOI, CHP, HP } , i, d, t
Pdev,i,d,t =
X
mds wdev,i,d,t,k · Pdev,i,d,t,k
∀dev ∈ {BOI, CHP, HP } , i, d, t
wdev,i,d,t,k
k
Part load of CFWHs and EHs: Q˙ dev,i,d,t ≤ Q˙ nom dev,i
(B.15)
∀dev ∈ {CF W H, EH} , i, d, t
(B.16)
Q˙ STC,i,d,t ≤ ηSTC,i,d,t · zSTC,i · ASTC,i · Id,t
940
941
942
(B.17)
∀i, d, t
(B.18)
AC C
PPV,i,d,t ≤ ηPV,i,d,t · zPV,i · APV,i · Id,t · η¯INV
939
∀i, d, t
Power generation of PV:
EP
938
(B.14)
∀dev ∈ {CF W H, EH} , i, d, t
TE
Heat output of STC:
(B.13)
D
Q˙ dev,i,d,t = Pdev,i,d,t 937
M AN U
ydev,i,d,t =
(B.12)
∀dev ∈ {BOI, CHP, HP } , i, d, t
k
X
SC
k
References
[1] Directive 2010/31/EU on the energy performance of buildings - 19 May 2010, Official Journal of the European Union L 153/13.
[2] Federal Ministry for Economic Affairs and Energy, Germany’s new energy
943
policy. Heading towards 2050 with secure, affordable and environmentally
944
sound energy (Apr. 2012).
41
ACCEPTED MANUSCRIPT
[3] M. M. Maghanki, B. Ghobadian, G. Najafi, R. J. Galogah, Micro combined
946
heat and power (MCHP) technologies and applications, Renewable and Sus-
947
tainable Energy Reviews 28 (2013) 510–524. doi:10.1016/j.rser.2013.07.053.
948
URL
949
S136403211300498X
RI PT
945
http://www.sciencedirect.com/science/article/pii/
[4] A. Peacock, M. Newborough, Impact of micro-CHP systems on domestic
951
sector CO2 emissions, Applied Thermal Engineering 25 (17-18) (2005)
952
2653–2676. doi:10.1016/j.applthermaleng.2005.03.015.
953
URL
954
S1359431105001018
SC
950
M AN U
http://www.sciencedirect.com/science/article/pii/
955
[5] D. Haeseldonckx, L. Peeters, L. Helsen, W. D’haeseleer, The impact of
956
thermal storage on the operational behaviour of residential chp facilities
957
and the overall co2 emissions, Renewable and Sustainable Energy Reviews
958
11 (6) (2007) 1227–1243. doi:10.1016/j.rser.2005.09.004.
959
URL
960
S1364032105001036
D
http://www.sciencedirect.com/science/article/pii/
[6] I. Staffell, D. Brett, N. Brandon, A. Hawkes, A review of domestic
962
heat pumps, Energy & Environmental Science 5 (11) (2012) 9291–9306.
963
doi:10.1039/C2EE22653G.
964
URL
965
c2ee22653g#!divAbstract
967
968
969
970
EP
http://pubs.rsc.org/en/Content/ArticleLanding/2012/EE/
[7] S. A. Kalogirou, Solar thermal collectors and applications, Progress
AC C
966
TE
961
in
Energy
and
Combustion
Science
30
(3)
(2004)
231–295.
doi:10.1016/j.pecs.2004.02.001. URL
http://www.sciencedirect.com/science/article/pii/
S0360128504000103
971
[8] B. Parida, S. Iniyan, R. Goic, A review of solar photovoltaic technolo-
972
gies, Renewable and Sustainable Energy Reviews 15 (2011) 1625–1636.
973
doi:10.1016/j.rser.2010.11.032. 42
ACCEPTED MANUSCRIPT
974
URL
975
S1364032110004016
RI PT
http://www.sciencedirect.com/science/article/pii/
976
[9] Y. Tian, C. Zhao, A review of solar collectors and thermal energy
977
storage in solar thermal applications, Applied Energy 104 (2013) 538–553.
978
doi:10.1016/j.apenergy.2012.11.051.
979
URL
980
S0306261912008549
SC
http://www.sciencedirect.com/science/article/pii/
[10] R. Luthander, J. Wid´en, D. Nilsson, J. Palm, Photovoltaic self-
982
consumption in buildings: A review, Applied Energy 142 (2015) 80–94.
983
doi:10.1016/j.apenergy.2014.12.028.
984
URL
985
S0306261914012859
986
M AN U
981
http://www.sciencedirect.com/science/article/pii/
[11] Act for the development of renewable energy sources (original German
987
title: Gesetz f¨ ur den Ausbau erneuerbarer Energien) (Jul. 2014).
988
URL
989
gesamt.pdf
D
http://www.gesetze-im-internet.de/bundesrecht/eeg_2014/
[12] Act on Combined Heat and Power Generation (original German title:
991
Gesetz f¨ ur die Erhaltung, die Modernisierung und den Ausbau der Kraft-
992
W¨ arme-Kopplung) (Dec. 2015).
993
URL https://www.gesetze-im-internet.de/bundesrecht/kwkg_2016/
994
gesamt.pdf
996
997
998
999
1000
EP
[13] Program Renewable Energies Storage (original German title:
AC C
995
TE
990
KfW-
Programm Erneuerbare Energien “Speicher”) (Mar. 2016). URL http://tinyurl.com/battery-subsidies
[14] Energy Tax Act (original German title: Energiesteuergesetz) (Dec. 2015). URL http://www.gesetze-im-internet.de/bundesrecht/energiestg/ gesamt.pdf
43
ACCEPTED MANUSCRIPT
[15] E. S. Barbieri, P. R. Spina, M. Venturini, Analysis of innovative micro-chp
1002
systems to meet household energy demands, Applied Energy 97 (2012)
1003
723–733. doi:10.1016/j.apenergy.2011.11.081.
1004
URL
1005
S0306261911007872
RI PT
1001
http://www.sciencedirect.com/science/article/pii/
[16] E. S. Barbieri, F. Melino, M. Morini, Influence of the thermal
1007
energy storage on the profitability of micro-chp systems for resi-
1008
dential building applications,
1009
doi:10.1016/j.apenergy.2012.01.001.
1010
URL
1011
S0306261912000050
Applied Energy 97 (2012) 714–722.
http://www.sciencedirect.com/science/article/pii/
[17] M. Bianchi,
M AN U
1012
SC
1006
A. D. Pascale,
F. Melino,
Performance analysis of
1013
an integrated chp system with thermal and electric energy stor-
1014
age for residential application, Applied Energy 112 (2013) 928–938.
1015
doi:10.1016/j.apenergy.2013.01.088.
1016
URL
1017
S0306261913001074
TE
D
http://www.sciencedirect.com/science/article/pii/
[18] Y. Yang, S. Zhang, Y. Xiao, Optimal design of distributed energy resource
1019
systems coupled with energy distribution networks, Energy 85 (2015)
1020
433–448. doi:10.1016/j.energy.2015.03.101.
1021
URL
1022
S0360544215004107
1024
1025
1026
http://www.sciencedirect.com/science/article/pii/
AC C
1023
EP
1018
[19] R. Loulou, U. Remne, A. Kanudia, A. Lehtila, G. Goldstein, Documentation for the TIMES Model - Part I, energy Technology Systems Analysis Programme (Apr. 2005). URL http://iea-etsap.org/docs/TIMESDoc-Intro.pdf
1027
[20] M. Stadler, DER-CAM Decision Support Tool for Decentralized En-
1028
ergy Systems, https://building-microgrid.lbl.gov/sites/default/
44
ACCEPTED MANUSCRIPT
files/projects/DER-CAMPresentation%2012%20May%202016.pdf (May
1030
2016).
1031
RI PT
1029
[21] M. B. Blarke, COMPOSE: Compare Options for Sustainable Energy.
1032
(2008).
1033
URL http://energyinteractive.net/COMPOSE.ashx
[22] M. B. Blarke, K. Yazawa, A. Shakouri, C. Carmo, Thermal battery
1035
with CO2 compression heat pump: Techno-economic optimization of a
1036
high-efficiency Smart Grid option for buildings, Energy and Buildings 50
1037
(2012) 128–138. doi:10.1016/j.enbuild.2012.03.029.
1038
URL
1039
S0378778812001776
M AN U
SC
1034
http://www.sciencedirect.com/science/article/pii/
1040
[23] E. Merkel, D. Fehrenbach, R. McKenna, W. Fichtner, Modelling decen-
1041
tralised heat supply: An application and methodological extension in
1042
TIMES, Energy 74 (2014) 592–605. doi:10.1016/j.energy.2014.06.060.
1043
URL
1044
S0360544214007592
D
http://www.sciencedirect.com/science/article/pii/
[24] K. A. Pruitt, R. J. Braun, A. M. Newman, Evaluating shortfalls in mixed-
1046
integer programming approaches for the optimal design and dispatch
1047
of distributed generation systems, Applied Energy 102 (2013) 386–398.
1048
doi:10.1016/j.apenergy.2012.07.030.
1049
URL
1050
S0306261912005521
1052
1053
1054
1055
1056
EP
http://www.sciencedirect.com/science/article/pii/
AC C
1051
TE
1045
[25] H. Ren, W. Gao, Y. Ruan, Optimal sizing for residential CHP system,
Applied
Thermal
Engineering
28
(5-6)
(2008)
514–523.
doi:10.1016/j.applthermaleng.2007.05.001. URL
http://www.sciencedirect.com/science/article/pii/
S1359431107001810 [26] K.-U. Klatt, W. Marquardt, Perspectives for process systems engineering
45
ACCEPTED MANUSCRIPT
1057
- personal views from academia and industry, Computers & Chemical En-
1058
gineering 33 (3) (2009) 536–550. doi:10.1016/j.compchemeng.2008.09.002.
1059
URL
1060
S0098135408001737
RI PT
http://www.sciencedirect.com/science/article/pii/
[27] G. Abdollahi, M. Meratizaman, Multi-objective approach in ther-
1062
moenvironomic optimization of a small-scale distributed CCHP system
1063
with risk analysis, Energy and Buildings 43 (11) (2011) 3144–3153.
1064
doi:10.1016/j.enbuild.2011.08.010.
1065
URL
1066
S0378778811003446
SC
1061
M AN U
http://www.sciencedirect.com/science/article/pii/
1067
[28] E. Barbieri, Y. Dai, M. Morini, M. Pinelli, P. Spina, P. Sun, R. Wang,
1068
Optimal sizing of a multi-source energy plant for power heat and cool-
1069
ing generation, Applied Thermal Engineering 71 (2) (2014) 736–750.
1070
doi:10.1016/j.applthermaleng.2013.11.022.
1071
URL
1072
S1359431113008119
D
http://www.sciencedirect.com/science/article/pii/
[29] B. Pickering, S. Ikeda, R. Choudhary, R. Ooka, Comparison of Metaheuris-
1074
tic and Linear Programming Models for the Purpose of Optimising Build-
1075
ing Energy Supply Operation Schedule, in: P. K. Heiselberg (Ed.), CLIMA
1076
2016 - proceedings of the 12th REHVA World Congress, Vol. 6, Aalborg
1077
University, Department of Civil Engineering, 2016, pp. 1–10.
1078
URL http://vbn.aau.dk/files/233775414/paper_529.pdf
1080
1081
1082
EP
AC C
1079
TE
1073
[30] P. Voll, Automated optimization based synthesis of distributed energy supply systems, Ph.d. thesis, RWTH Aachen University (2013). URL http://publications.rwth-aachen.de/record/228954
[31] A. Ashouri, S. S. Fux, M. J. Benz, L. Guzzella, Optimal design and
1083
operation of building services using mixed-integer linear programming
1084
techniques, Energy 59 (2013) 365–376. doi:10.1016/j.energy.2013.06.053.
46
ACCEPTED MANUSCRIPT
1085
URL
1086
S0360544213005525
RI PT
http://www.sciencedirect.com/science/article/pii/
1087
[32] E. Merkel, R. McKenna, W. Fichtner, Optimisation of the capacity and
1088
the dispatch of decentralised micro-chp systems: A case study for the uk,
1089
Applied Energy 140 (2015) 120–134. doi:10.1016/j.apenergy.2014.11.036.
1090
URL
1091
S0306261914011982
SC
http://www.sciencedirect.com/science/article/pii/
[33] M. Ameri, Z. Besharati, Optimal design and operation of district heating
1093
and cooling networks with CCHP systems in a residential complex, Energy
1094
and Buildings 110 (2016) 135–148. doi:10.1016/j.enbuild.2015.10.050.
1095
URL
1096
S0378778815303649
M AN U
1092
http://www.sciencedirect.com/science/article/pii/
1097
[34] P. Voll, C. Klaffke, M. Hennen, A. Bardow, Automated superstructure-
1098
based synthesis and optimization of distributed energy supply systems,
1099
Energy 50 (2013) 374–388. doi:10.1016/j.energy.2012.10.045.
1100
URL
1101
S0360544212008079
TE
D
http://www.sciencedirect.com/science/article/pii/
[35] E. Mehleri, H. Sarimveis, N. Markatos, L. Papageorgiou, A mathemat-
1103
ical programming approach for optimal design of distributed energy
1104
systems at the neighbourhood level, Energy 44 (1) (2012) 96–104.
1105
doi:10.1016/j.energy.2012.02.009.
1106
URL
1108
1109
1110
http://www.sciencedirect.com/science/article/pii/
AC C
1107
EP
1102
S036054421200103X
[36] E. Mehleri, H. Sarimveis, N. Markatos, L. Papageorgiou, Optimal design and operation of distributed energy systems:
Application
to Greek residential sector, Renewable Energy 51 (2013) 331–342.
1111
doi:10.1016/j.renene.2012.09.009.
1112
URL
1113
S0960148112005757
http://www.sciencedirect.com/science/article/pii/
47
ACCEPTED MANUSCRIPT
[37] H. Harb, J. Reinhardt, R. Streblow, D. M¨ uller, MIP approach for
1115
designing heating systems in residential buildings and neighbourhoods,
1116
Journal of Building Performance Simulation 9 (3) (2016) 316–330.
1117
doi:10.1080/19401493.2015.1051113.
1118
URL
1119
2015.1051113
RI PT
1114
http://www.tandfonline.com/doi/full/10.1080/19401493.
[38] M. A. Lozano, J. C. Ramos, L. M. Serra, Cost optimization of the design
1121
of CHCP (combined heat, cooling and power) systems under legal con-
1122
straints, Energy 35 (2) (2010) 794–805. doi:10.1016/j.energy.2009.08.022.
1123
URL
1124
S0360544209003624
SC
1120
M AN U
http://www.sciencedirect.com/science/article/pii/
1125
[39] D. Buoro, M. Casisi, P. Pinamonti, M. Reini, Optimal synthesis and
1126
operation of advanced energy supply systems for standard and do-
1127
motic home, Energy Conversion and Management 60 (2012) 96–105.
1128
doi:10.1016/j.enconman.2012.02.008.
1129
URL
1130
S0196890412000623 [40] R. Renaldi,
D
TE
1131
http://www.sciencedirect.com/science/article/pii/
A. Kiprakis,
D. Friedrich,
An optimisation frame-
1132
work for thermal energy storage integration in a residential heat
1133
pump heating system,
1134
doi:10.1016/j.apenergy.2016.02.067.
1135
URL
1136
S0306261916302045
1138
1139
EP
http://www.sciencedirect.com/science/article/pii/
AC C
1137
[41] T. of
their
Applied Energy 186 (3) (2017) 520–529.
Wakui,
residential operating
R.
Yokoyama,
Optimal
structural
design
cogeneration
systems
in
consideration
restrictions,
Energy
64
(2014)
of
719–733.
1140
doi:http://www.sciencedirect.com/science/article/pii/S0360544213008323.
1141
URL 10.1016/j.energy.2013.10.002
1142
[42] T. Wakui, H. Kawayoshi, R. Yokoyama, Optimal structural design of 48
ACCEPTED MANUSCRIPT
residential power and heat supply devices in consideration of operational
1144
and capital recovery constraints, Applied Energy 163 (2016) 118–133.
1145
doi:10.1016/j.apenergy.2015.10.154.
1146
URL
1147
S0306261915013963
RI PT
1143
http://www.sciencedirect.com/science/article/pii/
[43] H. D. Vinod, Integer Programming and the Theory of Grouping, Jour-
1149
nal of the American Statistical Association 64 (326) (1969) 506–519.
1150
doi:10.2307/2283635.
1151
URL http://www.jstor.org/stable/2283635 [44] F.
Dom´ınguez-Mu˜ noz, Gallardo-Salazar,
J.
M AN U
1152
SC
1148
M.
1153
M.
Selection
1154
CHP optimization,
1155
doi:10.1016/j.enbuild.2011.07.024.
1156
URL
1157
S037877881100329X
Cejudo-L´opez, of
typical
A.
Carrillo-Andr´es,
demand
days
for
Energy and Buildings 43 (2011) 3036–3043.
http://www.sciencedirect.com/science/article/pii/
[45] T. Sch¨ utz, M. Schraven, H. Harb, M. Fuchs, D. M¨ uller, Clustering algo-
1159
rithms for the selection of typical demand days for the optimal design of
1160
building energy systems, in: Proceedings of ECOS 2016 - The 29th In-
1161
ternational Conference on Efficiency, Cost, Optimization, Simulation and
1162
Environmental Impact of Energy Systems, University of Ljubljana, Faculty
1163
of Mechanical Engineering, 2016, pp. 1–12.
1165
1166
1167
1168
TE
EP
[46] Association of German Engineers, Economic efficiency of building instal-
AC C
1164
D
1158
lations - Fundamentals and economic calculation, Guideline VDI 2067-1, Beuth Verlag GmbH, D¨ usseldorf (Sep. 2012).
[47] T. Sch¨ utz, H. Harb, M. Fuchs, D. M¨ uller, Optimal Design of Building Energy Systems for Residential Buildings, in: P. K. Heiselberg (Ed.), CLIMA
1169
2016 - proceedings of the 12th REHVA World Congress, Vol. 6, Aalborg
1170
University, Department of Civil Engineering, 2016, pp. 1–10.
1171
URL http://vbn.aau.dk/files/233774153/paper_157.pdf 49
ACCEPTED MANUSCRIPT
[48] T. Sch¨ utz, L. Schiffer, H. Harb, M. Fuchs, D. M¨ uller, Optimal design of
1173
energy conversion units and envelopes for residential building retrofits
1174
using a comprehensive MILP model, Applied Energy 185 (2017) 1–15.
1175
doi:10.1016/j.apenergy.2016.10.049.
1176
URL
1177
S0306261916314933
http://www.sciencedirect.com/science/article/pii/
[49] Arbeitsgemeinschaft f¨ ur sparsamen und umweltfreundlichen Energiever-
SC
1178
RI PT
1172
brauch (ASUE), Bhkw-kenndaten 2014/15 (Oct. 2014).
1180
URL http://www.asue.de/blockheizkraftwerke/broschueren/05_10_
1181
14_bhkw-kenndaten_2014-15
1182
M AN U
1179
[50] Bundesnetzagentur, EEG-Umlage. Was ist die EEG-Umlage und wie
1183
funktioniert sie? (Oct. 2016).
1184
URL
1185
Sachgebiete/Energie/Verbraucher/Energielexikon/EEGUmlage.html
1188
& Sons, Inc., 2013.
D
1187
[51] H. P. Williams, Model Building in Mathematical Programming, John Wiley
[52] Bundesnetzagentur, Photovoltaikanlagen - Datenmeldungen und EEG-
TE
1186
https://www.bundesnetzagentur.de/SharedDocs/FAQs/DE/
1189
Verg¨ utungss¨ atze (May 2017).
1190
URL
1191
ElektrizitaetundGas/Unternehmen_Institutionen/
1192
ErneuerbareEnergien/Photovoltaik/DatenMeldgn_EEG-VergSaetze/
1193
DatenMeldgn_EEG-VergSaetze_node.html
1195
1196
1197
1198
EP
AC C
1194
https://www.bundesnetzagentur.de/DE/Sachgebiete/
[53] EEX, KWK Index (Jun. 2017). URL
https://www.eex.com/en/market-data/power/spot-market/
kwk-index
[54] J. A. Duffie, W. A. Beckman, Solar Engineering of Thermal Processes, 4th Edition, John Wiley & Sons, Inc., 2013.
50
ACCEPTED MANUSCRIPT
[55] R. Perez, P. Ineichen, R. Seals, J. Michalsky, R. Stewart, Modeling daylight
1200
availability and irradiance components from direct and global irradiance,
1201
Solar Energy 44 (1990) 271–289. doi:10.1016/0038-092X(90)90055-H.
1202
URL
1203
0038092X9090055H
1204
RI PT
1199
http://www.sciencedirect.com/science/article/pii/
[56] S. Dubey,
J. N. Sarvaiya,
B. Seshadri,
Temperature Dependent
Photovoltaic (PV) Efficiency and Its Effect on PV Production
1206
in the World - A Review, Energy Procedia 33 (2013) 311–321.
1207
doi:10.1016/j.egypro.2013.05.072.
1208
URL
1209
S1876610213000829
http://www.sciencedirect.com/science/article/pii/
M AN U
1210
SC
1205
[57] United Nations, Adoption of the Paris Agreement (2015).
1211
URL http://unfccc.int/resource/docs/2015/cop21/eng/l09r01.pdf
1212
[58] Umweltbundesamt, Entwicklung der spezifischen Kohlendioxid-Emissionen des deutschen Strommix in den Jahren 1990 - 2016 (May 2017).
1214
URL
1215
medien/1410/publikationen/2017-05-22_climate-change_15-2017_
1216
strommix.pdf
D
1213
TE
https://www.umweltbundesamt.de/sites/default/files/
[59] L. A. Wolsey, Integer Programming, Vol. 42, Wiley New York, 1998.
1218
[60] T. Wakui, R. Yokoyama, Optimal structural design of residential co-
1219
generation systems with battery based on improved solution method
1220
for mixed-integer linear programming, Energy 84 (2015) 106–120.
1222
1223
1224
AC C
1221
EP
1217
doi:10.1016/j.energy.2015.02.056. URL
http://www.sciencedirect.com/science/article/pii/
S036054421500211X
[61] Bundesamt
1225
vironment
1226
Aktualisierte
f¨ ur
Bauwesen
Consulting und
und
Potsdam
erweiterte
Raumordnung, GmbH,
Deutscher
Testreferenzjahre
51
Climate
von
&
En-
Wetterdienst, Deutschland
ACCEPTED MANUSCRIPT
1227
f¨ ur
mittlere,
extreme
und
zuk¨ unftige
Witterungsverh¨altnisse,
1228
URL
1229
Regelungen/Testreferenzjahre/Testreferenzjahre/01_start.html?
1230
nn=739044¬First=true&docId=743442 [accessed 24.01.2017] (Jul.
1231
2011).
RI PT
http://www.bbsr-energieeinsparung.de/EnEVPortal/DE/
[62] D. M¨ uller, M. Lauster, A. Constantin, M. Fuchs, P. Remmen, AIXLIB
1233
- AN OPEN-SOURCE MODELICA LIBRARY WITHIN THE IEA-EBC
1234
ANNEX 60 FRAMEWORK, in: BauSIM 2016, 2016, pp. 3–9.
1235
URL http://www.iea-annex60.org/downloads/2016-bausim-aixlib.
1236
pdf [63] P. Remmen,
M. Lauster,
M AN U
1237
SC
1232
M. Mans,
M. Fuchs,
T. Osterhage,
1238
D. M¨ uller, TEASER: an open tool for urban energy modelling of
1239
building stocks, Journal of Building Performance Simulation (2017)
1240
1–15doi:10.1080/19401493.2017.1283539.
1241
URL
1242
2017.1283539
D
[64] I. Richardson, M. Thomson, D. Infield, C. Clifford, Domestic electricity
TE
1243
http://www.tandfonline.com/doi/full/10.1080/19401493.
1244
use:
1245
42 (10) (2010) 1878–1887. doi:10.1016/j.enbuild.2010.05.023.
1246
URL
1247
S0378778810001854
1249
1250
1251
1252
EP
http://www.sciencedirect.com/science/article/pii/
[65] IEA Energy Conservation in Buildings & Community Systems, Annex 42
AC C
1248
A high-resolution energy demand model, Energy and Buildings
The Simulation of Building-Integrated Fuel Cell and Other Cogeneration Systems (COGEN-SIM), accessed May 11th 2016 (2008). URL http://www.ecbcs.org/annexes/annex42.htm
[66] M. Chen, G. A. Rinc´ on-Mora, Accurate electrical battery model capable
1253
of predicting runtime and i-v performance, IEEE Transactions on Energy
1254
Conversion 21 (2) (2006) 504–511. doi:10.1109/TEC.2006.874229.
1255
URL http://ieeexplore.ieee.org/abstract/document/1634598/ 52
ACCEPTED MANUSCRIPT
1257
1258
[67] STAWAG, accessed June 21st 2017 (2017). [link]. URL https://www.stawag.de/ [68] Umweltbundesamt,
Verhindern Herkunftsnachweise ”Greenwashing”?,
1259
accessed June 20th 2017 (Aug. 2013).
1260
URL
1261
verhindern-herkunftsnachweise-greenwashing
https://www.umweltbundesamt.de/service/uba-fragen/
¨ [69] Umweltbundesamt, Marktanalyse Okostrom (Apr. 2014).
SC
1262
RI PT
1256
1263
URL
1264
medien/376/publikationen/texte_04_2014_marktanalyse_
1265
oekostrom_0.pdf
M AN U
1266
https://www.umweltbundesamt.de/sites/default/files/
[70] MONTANA Energie-Handel, MONTANA Biogas, accessed June 21st 2017
1267
(Jun. 2017).
1268
URL
1269
montana-erdgas/tarife/biogas/
[71] Statistisches Bundesamt, Preise, daten zur energiepreisentwicklung (May
D
1270
https://www.montana-energie.de/privatkunden/erdgas/
1271
2017).
1272
URL
1273
Preise/Energiepreise/EnergiepreisentwicklungPDF_5619001.pdf?
1274
__blob=publicationFile
TE
EP
1275
https://www.destatis.de/DE/Publikationen/Thematisch/
[72] Gurobi Optimization, Linear Programming (LP) - A Primer on the Basics, accessed June 26th 2017 (2017).
1277
URL http://www.gurobi.com/resources/getting-started/lp-basics
1278
1279
1280
AC C
1276
[73] H. Wolisz, T. Sch¨ utz, T. Blanke, M. Hagenkamp, M. Kohrn, M. Wesseling, D. M¨ uller, Cost optimal sizing of smart buildings’ energy system components considering changing end-consumer electricity markets,
1281
Energydoi:10.1016/j.energy.2017.06.025.
1282
URL
1283
S0360544217310198
http://www.sciencedirect.com/science/article/pii/
53
ACCEPTED MANUSCRIPT
Table C.1: Variables and parameters
RI PT
Appendix C. Tables and Figures
Symbol
Description
Unit
A
Area
CRF
Capital Recovery Factor
E˙
Gas consumption
El
Annual el. consumption
Etar
Annual el. consumption limits of tariffs
MWh
G
Annual gas consumption of heating devices
MWh
Gtar
Annual gas consumption limits of tariffs
MWh
I
Solar irradiation
kW
M
Big-M, upper bound for a specific variable
—
Q˙
Heat flow rate
kW
P
Electrical power
kW
S
Storage’s state of charge
m2
a−1
M AN U
SC
kW
Temperature
V
Volume
TE
D
T
b
MWh
% ◦
C
m3
Price dynamic cash value
—
Costs
Euro
Storage capacity
kWh
Storage charging power
kW
Storage discharging power
kW
AC C e
Revenue
Euro
emi
Emissions
kg
f
Fixed parameters
—
k1
STC linear loss factors
W/(m2 ·K)
k2
STC quadratic loss factors
10−3 ·W/(m2 ·K2 )
p
Subsidy and remuneration rate
—
c cap ch dch
EP
1284
Continued on next page 54
ACCEPTED MANUSCRIPT
Table C.1: Variables and parameters
Description
rv
Residual value of each type of device
sub
Subsidies
t
Time period
tar
(Binary) decision on tariff selection
tax
Tax
w
Weighting variable
x
(Binary) decision of purchase
y
(Binary) activation of heating devices
z
Number of installed solar modules
Unit
RI PT
Symbol
—
Euro h
—
SC
—
—
—
M AN U
— —
Table C.2: Greek letters
Unit
Difference
—
TE
∆
PV loss factor
%/K
δ
(Binary) decision on KWKG subsidy
—
ζ
Linearized product of two variables
—
EP
γ
Description
D
Symbol
Efficiency
%
κ
Heat capacity
J / (kg·K)
ρ
Density
kg / m3
ϕ
Storage’s loss factor
—
AC C
η
55
ACCEPTED MANUSCRIPT
Table C.3: Subscripts and abbreviations
Description
BAT
Battery
BES
Building Energy System
BOI
Boiler
CFWH
Continuous flow water heater
CHP
Combined Heat and Power
COP
Coefficient of Performance
DHL
Design Heat Load
DHW
Domestic Hot Water
EEG
German Renewable Energy Sources Act
EEX
European Energy Exchange
EH
Electrical resistance Heater
EM
Electricity mix
SC
M AN U
HP
Heat pump
International Energy Agency
D
IEA INV
Inverter
German Reconstruction Credit Institute
TE
KfW
RI PT
Symbol
KWKG
German Act on Combined Heat and Power Generation
LB
Lower bound
Mixed-Integer Linear Program
MINLP
Mixed-Integer Nonlinear Program
PL
House’s plug loads
AC C
EP
MILP
PV
Photovoltaic
RES
Renewable Energy System
SH
Space Heating
SOS2
Special Ordered Set of Type 2
STC
Solar Thermal Collector
Continued on next page
56
ACCEPTED MANUSCRIPT
Description
TES
Thermal Energy Storage
UB
Upper bound
VDI
Association of German Engineers
ann
Annualized
cap
Capacity
d
Typical demand day
dem
Demand
dev
Device
M AN U
eeg
EEG levy
el
Electricity
exp
Export
feed
Feed-in
fix
Fixed price Gas
D
gas hpt
Heat pump tariff
Counting index, type index
TE
i ∗
i
Set of small scale micro CHP units
i∗
Set of large scale micro CHP units Import
infl
Inflation
inv
Investment
AC C
EP
imp
j
j
SC
Symbol
RI PT
Table C.3: Subscripts and abbreviations
∗
Tariff index Subset of non-hpt electricity tariffs
k
Counting index, data sheet interpolation
lvl
Tariff level
max
Maximum
Continued on next page
57
ACCEPTED MANUSCRIPT
Description
mds
Manufacturer data sheet
met
Metering
nom
Nominal
o&m
Operation and maintenance
sell
Sold to the distribution grid
spec
Specific
sub
Subsidies
var
Variable price
M AN U
SC
Symbol
RI PT
Table C.3: Subscripts and abbreviations
Table C.4: Summary of considered German subsidies and market characteristics
Implication
EEG (feed-in limit)
Max. 70% of PV peak power can be fed into the grid
46-49
EEG (levy)
40% of the EEG levy has to be paid on self-consumed electricity
18-24
D
Market characteristic / Subsidy
Equations
EEG (feed-in remuneration)
TE
This only applies to facilities with more than 10 kW capacity Special feed-in remuneration for electricity from PV
25-29
EP
Special rates for different installed capacities
KWKG
Financial support for battery assisted PV systems
38-42
However, max. 50% of PV peak power can be fed into the grid
46-49
Variable payment for self-consumption and feed-in from CHPs
31-37
AC C
KfW 275
Alternatively fixed payment for small-scale units
EStG HP tariffs
Gas and electricity tariffs
Fuel tax exemption for CHP units
9
Option for cheaper electricity tariff for HPs
11
Multiple tariffs with different fixed and variable costs
2-17
Tiered pricing structure Different emission factors for each tariff
58
67-68
RI PT
ACCEPTED MANUSCRIPT
Table C.5: Summary of investigated buildings
Type
Location
Apartments
Residents
Floor area in m
2
Roof area in m
SFH
Bavaria
1
3
121
40
2
MFH
Hamburg
8
19
520
100
3
AB
Berlin
15
34
1005
170
SH demand
Electricity demand
in kWh/a
in kWh/a
in kWh/a
950
6050
3750
8550
26035
18700
15450
50315
33950
M AN U
1
DHW demand
2
SC
Number
Type
BOI
1
E˙ mds
cinv
in kW
in kW
in kW
in Euro
—
2.8 ; 4.2 ; 15.8
1,970
4.1 ; 6.1 ; 20.3
—
4.1 ; 6.2 ; 23.2
2,120
3
5.5 ; 8.3 ; 27.7
—
5.6 ; 8.5 ; 31.7
2,305
4
7.4 ; 11.0 ; 36.8
—
7.4 ; 11.3 ; 42.1
2,680
1
2.5
1.0
4.2
9,585
2
4.7 ; 8.0
1.5 ; 3.0
7.7 ; 13.3
15,853
3
4.7 ; 12.5
1.5 ; 4.7
7.7 ; 21.1
19,472
4
12.3 ; 16.1 ; 20.1
4.3 ; 6.4 ; 8.5
20.3 ; 26.4 ; 33.5
25,542
5
24.0 ; 36.0 ; 45.0
10.0 ; 15.0 ; 20.0
45.5 ; 63.3 ; 79.9
38,002
EP
AC C
P mds
2.8 ; 4.1 ; 13.8
2
CHP
Q˙ mds
TE
Component
D
Table C.6: Available gas-fired heat generators.
59
ACCEPTED MANUSCRIPT
P nom
cinv
in kW
in kW
in Euro
CFWH / EH
1
2.0
2.0
179
2
6.0
6.0
199
3
8.0
8.0
4
12.0
12.0
HP
1
5.0
1.3
2 3
209
219
4,390
8.0
2.1
4,990
11.0
2.9
6,548
15.0
3.9
7,460
D
4
SC
Q˙ nom
Type
M AN U
Component
RI PT
Table C.7: Available electrical heat generators.
Table C.8: Scaled characteristics of the HP units.
(Scaled) Q˙ mds / (scaled) P mds
in ◦ C
—
TE
Ambient temperature
0.32 / 0.17
0.32 / 0.17
0.09 / 0.04
-15
0.57 / 0.27
0.51 / 0.22
0.12 / 0.05
-7
0.61 / 0.26
0.55 / 0.22
0.13 / 0.05
-3
0.69 / 0.26
0.62 / 0.22
0.14 / 0.05
0
0.74 / 0.26
0.66 / 0.22
0.15 / 0.05
2
0.79 / 0.26
0.70 / 0.23
0.16 / 0.05
7
1.08 / 0.28
0.96 / 0.23
0.21 / 0.05
10
1.17 / 0.28
1.05 / 0.24
0.23 / 0.05
20
1.48 / 0.27
1.33 / 0.23
0.29 / 0.05
AC C
EP
-20
60
Table C.9: Available solar thermal collectors.
Type
RI PT
ACCEPTED MANUSCRIPT
cinv
A
η0
k1
k2
in m2
—
in W/(m2 ·K)
in 10−3 ·W/(m2 ·K2 )
2.00
0.80
3.24
11.7
2
0.90
0.73
3.47
8.0
3
2.32
0.82
3.33
440
SC
1
in Euro
M AN U
23.0
Table C.10: Available solar power generators and inverters.
AC C
η
γ
cinv
in m2
in kW
—
%/K
in Euro
1.73
0.36
0.21
0.30
539
2
1.71
0.30
0.17
0.41
224
3
1.64
0.27
0.16
0.42
184
1
—
2.4
0.98
—
703
2
—
5.1
0.97
—
1,040
3
—
8.8
0.97
—
1,358
4
—
10.0
0.98
—
1,491
5
—
15.6
0.98
—
1,661
6
—
21.2
0.98
—
1,812
7
—
28.0
0.98
—
2,563
8
—
36.0
0.98
—
3,716
EP
INV
1
P nom
A
D
PV
Type
TE
Component
61
190 299
SC
RI PT
ACCEPTED MANUSCRIPT
Table C.11: Available storage units.
Type
cap
dch,max PBAT
cinv
in kWh/24h
—
kW
kW
in Euro
1
6.4
—
0.96
2.0
2.0
5,098
2
8.0
—
0.96
3.0
3.0
5,620
3
11.0
—
0.96
3.7
3.7
6,679
4
16.0
—
0.96
3.7
3.7
7,979
5
22.0
—
0.96
3.7
3.7
9,129
1
0.12
1.7
—
—
—
389
0.49
2.2
—
—
—
650
0.75
4.5
—
—
—
769
0.98
3.2
—
—
—
980
5
1.50
4.1
—
—
—
1,380
6
2.00
4.6
—
—
—
1,840
2 3
AC C
EP
4
TE
TES
ch,max PBAT
D
in kWh | m3 BAT
η
ϕ
M AN U
Component
62
RI PT
ACCEPTED MANUSCRIPT
EtarUB
cvar
cfix
espec
in MWh/a
in MWh/a
in Euro/kWh
in Euro/kWh
in kg/kWh
Standard el.
1
0.00
2.80
0.2713
73.02
0.527
2
2.80
6.00
0.2699
77.02
0.527
3
6.00
100.00
0.2687
84.16
0.527
Eco el.
1
0.00
2.80
0.2802
73.02
0.332
2
2.80
6.00
0.2735
92.02
0.332
3
6.00
9.00
0.2723
99.16
0.332
4
9.00
12.00
0.2713
107.73
0.332
12.00
100.00
0.2705
117.73
0.332
0.00
100.00
0.2013
91.51
0.527
1
0.00
5.38
0.0798
39.98
0.250
2
5.38
12.28
0.0615
138.66
0.250
3
12.28
100.00
0.0580
182.50
0.250
1
0.00
5.38
0.0858
39.98
0.225
2
5.38
12.28
0.0675
138.66
0.225
3
12.28
100.00
0.0640
182.50
0.225
5
AC C
Eco gas
EP
Standard gas
1
TE
Heat pump el.
M AN U
EtarLB
Level
D
Tariff
SC
Table C.12: Available electricity and gas tariffs.
63
ACCEPTED MANUSCRIPT
Value
Unit
10
a
5
%
Price change electricity
1.0512
1/a
Price change gas
1.0074
1/a
Price change EEX compensation
0.9551
1/a
15
a
Observation period Interest rate
M AN U
Lifetime CHP Lifetime boiler Lifetime EH Lifetime CFWH Lifetime HP Lifetime PV Lifetime STC
D
Lifetime TES Lifetime BAT
SC
Parameter
RI PT
Table C.13: Economic parameters.
a
20
a
20
a
18
a
20
a
20
a
20
a
15
a
15
a
EP
TE
Lifetime INV
20
AC C
Table C.14: Model constraints and variables.
New model
Original model
No subsidies
21993
16747
21948
1560
1560
1560
26134
24285
26108
Number integer variables
9
9
9
Number binary variables
2952
2943
2942
Number constrains
Number SOS constraints Number variables
64
RI PT
ACCEPTED MANUSCRIPT
Cost minimization Original
No subsidies
New model
M AN U
New model
SC
Table C.15: Results, single family house
Opt.
Rec.
20% CO2 reduction Original
No subsidies
Opt.
Rec.
1879.75
1870.60
1877.94
2158.01
1879.75
1870.60
1877.94
2165.21
Total emissions in tCO2 /a
-0.100
-0.078
-0.099
2.921
-0.100
-0.078
-0.099
2.248
Total investments in Euro/a
781.46
768.80
779.58
409.51
781.46
768.80
779.58
530.29
TES investments in Euro/a
58.34
45.68
56.46
34.91
58.34
45.68
56.46
69.20
Demand costs in Euro/a
1459.14
1464.80
1459.53
1488.57
1459.14
1464.80
1459.53
1476.35
Metering costs in Euro/a
215.68
215.68
215.68
215.68
215.68
215.68
215.68
117.00
Revenues in Euro/a
670.66
670.66
670.66
19.55
670.66
670.66
670.66
37.79
BOI in kW
13.8
EH in kW
0
3
0.49
PV in m2
37.62
2
STC in m
TE
13.8
13.8
13.8
13.8
13.8
13.8
0
0
2
0
0
0
2
0.33
0.49
0.12
0.49
0.33
0.49
0.75
37.62
37.62
8.18
37.62
37.62
37.62
11.45
2.32
2.32
2.32
2.32
2.32
2.32
2.32
6.96
660.8
1340.7
203.4
753.3
606.2
1340.7
203.4
851.1
AC C
Run times in s
13.8
EP
TES in m
D
Total costs in Euro/a
65
RI PT
ACCEPTED MANUSCRIPT
Table C.16: Results, multi family house
Cost minimization Original
No subsidies
Opt.
Rec.
No subsidies
Opt.
Rec.
7313.96
7318.92
7818.05
7229.53
7214.07
7216.87
Total emissions in tCO2 /a
12.449
12.525
12.528
11.494
8.964
9.036
9.042
9.184
Total investments in Euro/a
3892.94
3891.7
3895.31
2411.75
4322.08
4318.31
4324.45
3951.58
TES investments in Euro/a
87.96
86.72
90.33
69.20
87.96
84.19
90.33
69.20
Demand costs in Euro/a
4105.34
4106.32
4105.66
4645.83
3854.36
3850.29
3849.19
3556.4
Metering costs in Euro/a
259.52
255.52
255.52
266.66
255.52
255.52
255.52
259.52
EEG levy in Euro/a
0
0
0
0
459.83
462.09
462.41
0
Revenues in Euro/a
675.95
681.16
682.08
24.74
1245.61
1245.06
1246.96
175.85
Subsidies in Euro/a
440.92
446.79
446.63
0
445.40
451.54
451.10
0
BOI in kW
0
CHP in kW
12.5
EH in kW
12
TE
D
M AN U
7325.72
Original
Total costs in Euro/a
0
0
20.3
0
0
0
20.3
12.5
12.5
0
12.5
12.5
12.5
2.5
12
12
8
12
12
12
0
EP
BAT in kWh
7526.96
New model
SC
New model
20% CO2 reduction
0
0
0
22
0
0
0
22
0.98
0.93
0.98
0.75
0.98
0.89
0.98
0.75
PV in m2
60.52
60.52
60.52
99.78
99.78
99.78
99.78
89.97
2
0
0
0
0
0
0
0
9.28
Run times in s
2935.7
1969.8
183.2
1449.9
1581.0
710.5
32.0
1777.6
3
AC C
TES in m
STC in m
66
Table C.17: Results, apartment building
Cost minimization No subsidies
Opt.
Rec.
New model
SC
Original
20% CO2 reduction Original
No subsidies
Opt.
Rec.
11132.50
11120.01
11127.67
11715.65
11400.85
11405.32
11415.20
11774.60
Total emissions in tCO2 /a
15.432
15.547
15.566
21.016
12.346
12.438
12.438
16.813
Total investments in Euro/a
5305.27
5297.73
5307.69
5334.66
6304.78
6300.52
6300.47
5808.07
TES investments in Euro/a
123.86
116.32
126.28
87.96
165.14
150.56
160.84
87.96
Demand costs in Euro/a
8338.83
8347.89
8344.1
6046.94
7552.32
7496.45
7605.69
5809.54
Metering costs in Euro/a
266.66
266.66
266.66
259.52
274.52
274.52
274.52
255.52
EEG levy in Euro/a
747.07
750.48
751.46
0
794.60
795.07
796.87
0
Revenues in Euro/a
2777.07
2784.86
2784.86
189.53
2656.01
2607.03
2656.06
401.95
Subsidies in Euro/a
986.93
995.38
996.58
0
1118.63
1102.81
1159.99
0
BOI in kW
13.8
13.8
13.8
20.3
13.8
13.8
13.8
20.3
CHP in kW
12.5
12.5
12.5
8
20.1
20.1
20.1
8
EH in kW
12
12
12
12
6
6
6
12
BAT in kWh
0
0
0
22
0
0
0
22
1.50
1.36
1.50
0.98
2.00
1.85
2.00
0.98
168.49
168.49
168.49
127.59
166.85
165.22
166.85
166.85
PV in m2
D
TE
EP
TES in m3
M AN U
Total costs in Euro/a
AC C
New model
RI PT
ACCEPTED MANUSCRIPT
STC in m2
0
0
0
2.32
2.32
4.64
2.32
2.32
El. tariff
std.
std.
std.
std.
eco
eco
eco
std.
Run times in s
3283.0
3136.1
0.4
528.4
4613.0
4842.2
78.3
1275.5
67
Table C.18: Sensitivity analysis
Opt.
+5% demands
Rec.
Opt.
-5% tariffs
Rec.
Single family house
Opt.
SC
-5% demands
RI PT
ACCEPTED MANUSCRIPT
+5% tariffs
Rec.
Opt.
Rec.
1802.71
1811.01
1959.11
1959.02
1806.83
1806.69
1952.88
1952.60
BOI in kW
13.8
13.8
13.8
13.8
13.8
13.8
13.8
13.8
EH in kW
2
2
0
0
0
0
0
0
TES in m3
0.12
0.49
0.49
0.49
0.49
0.49
0.49
0.49
M AN U
Total costs in Euro/a
Multi family house 6904.51
7007.99
7453.44
7452.74
6959.55
7007.18
7450.71
7450.40
BOI in kW
27.7
0
0
0
27.7
0
0
0
CHP in kW
0
12.5
12.5
12.5
0
12.5
12.5
12.5
EH in kW
0
12
12
12
0
12
12
12
3
0.49
0.98
0.98
0.98
0.49
0.98
0.98
0.98
PV in m2
93.24
60.52
60.52
60.52
94.88
60.52
60.52
60.52
2
4.64
0
0
0
4.64
0
0
0
TE
STC in m
EP
TES in m
D
Total costs in Euro/a
Apartment building
10612.79
10609.87
11631.12
11688.43
10688.52
10683.92
11582.37
11582.68
BOI in kW
13.8
13.8
20.3
13.8
13.8
13.8
13.8
13.8
AC C
Total costs in Euro/a
CHP in kW
12.5
12.5
20.1
12.5
12.5
12.5
12.5
12.5
EH in kW
12
12
0
12
12
12
12
12
3
1.50
1.50
2.00
1.50
1.50
1.50
1.50
1.50
TES in m
68
5
RI PT
Continuous Discrete
SC
40 35 30 25 20 15 10 5 0 0
10
15
M AN U
Investment costs in 1000 EUR
ACCEPTED MANUSCRIPT
Nominal electricity output in kW
20
Figure C.1: Continuous and discrete approaches for CHP units (single column figure)
15 10
D TE
20
Constant efficiency No part load threshold Piecewise linearization
EP
Electrical efficiency in %
25
5
AC C
0 0
20
40
60
Modulation level in %
80
100
Figure C.2: Part load models for one exemplary CHP unit (single column figure)
69
−
Electrical loads
DHW CFWH
Boiler CHP EH HP
−
TES
BAT
+
M AN U
Natural gas
SC
SH
~
RI PT
ACCEPTED MANUSCRIPT
Electrical grid
AC C
EP
TE
D
Figure C.3: Building energy system structure (single column figure)
Figure C.4: Building’s heat balance (single column figure)
70
EP
TE
D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
Figure C.5: Building’s electricity balance (single column figure)
71
ACCEPTED MANUSCRIPT Table C.4: Summary of considered German subsidies and market characteristics
EEG (levy) EEG (feed-in remuneration) KfW 275
KWKG
Max. 70% of PV peak power can be fed into the grid 40% of the EEG levy has to be paid on self-consumed electricity This only applies to facilities with more than 10 kW capacity Special feed-in remuneration for electricity from PV Special rates for different installed capacities Financial support for battery assisted PV systems However, max. 50% of PV peak power can be fed into the grid Variable payment for self-consumption and feed-in from CHPs Alternatively fixed payment for small-scale units Fuel tax exemption for CHP units Option for cheaper electricity tariff for HPs Multiple tariffs with different fixed and variable costs Tiered pricing structure Different emission factors for each tariff
Equations 46-49 18-24
25-29 38-42 46-49 31-37
Citation in the text (lines 237-239):
9 11 2-17 67-68
M AN U
SC
EStG HP tariffs Gas and electricity tariffs
Implication
RI PT
Market characteristic / Subsidy EEG (feed-in limit)
Latex version:
TE D
A main novelty of this paper is the consideration of many German subsidies and market characteristics. Table C.4 summarizes the implemented characteristics and links them to the corresponding equations used in this paper.
AC C
EP
\begin{table}[h!] \caption{Summary of considered German subsidies and market characteristics} \centering \hspace*{-10em} \begin{tabular}[l]{@{}lll} \hline Market characteristic / Subsidy & Implication & Equations\\ \hline EEG (feed-in limit) & Max. 70\% of PV peak power can be fed into the grid & \ref{eqn:limit feed in}-\ref{eqn:end_limit_feed_in}\\ EEG (levy) & 40\% of the EEG levy has to be paid on self-consumed electricity & \ref{eqn: eeg more than 10 kW installed_1}-\ref{eqn:eeg_levy_end}\\ & This only applies to facilities with more than 10 kW capacity & \\ EEG (feed-in remuneration) & Special feed-in remuneration for electricity from PV & \ref{eqn:rev_feed_in_pv_1}-\ref{eqn:rev_feed_in_pv_ende}\\ & Special rates for different installed capacities & \\ \hline KfW 275 & Financial support for battery assisted PV systems & \ref{eqn: sub_bat_start}-\ref{eqn: sub_bat_end}\\ & However, max. 50\% of PV peak power can be fed into the grid & \ref{eqn:limit feed in}-\ref{eqn:end_limit_feed_in}\\ \hline
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
KWKG & Variable payment for self-consumption and feed-in from CHPs & \ref{eqn:subs_CHP_large}-\ref{eqn:kwkg_ende} \\ & Alternatively fixed payment for small-scale units & \\ \hline EStG & Fuel tax exemption for CHP units & \ref{eqn:cost_gas_energy_tax_chp}\\ \hline HP tariffs & Option for cheaper electricity tariff for HPs & \ref{eqn: allow_hp_tariff} \\ \hline Gas and electricity tariffs & Multiple tariffs with different fixed and variable costs & \ref{eqn:tariff_start}-\ref{eqn:tariff_metering} \\ & Tiered pricing structure & \\ & Different emission factors for each tariff & \ref{eqn:co2_gas}-\ref{eqn:co2_el}\\ \hline \end{tabular} \label{tab: market characteristics} \end{table}
ACCEPTED MANUSCRIPT Highlights:
EP
TE D
M AN U
SC
RI PT
MILP with discrete, device-specific storage modeling Consideration of German subsidies, market characteristics and multiple tariffs Storage model is more accurate but marginally affects the calculating times Subsidies and market characteristics strongly influence the optimal energy systems Results are robust to variations in energy tariff costs and demands
AC C
1. 2. 3. 4. 5.