optimal operation of heat supply systems with piping network
TRANSCRIPT
Optimal operation of heat supply systems withpiping network
著者 Yokoyama Ryohei, Kitano Hiroyuki, WakuiTetsuya
journal orpublication title
Energy
volume 137page range 888-897year 2017-10-15権利 (c) 2017. This manuscript version is made
available under the CC-BY-NC-ND 4.0 licensehttp://creativecommons.org/licenses/by-nc-nd/4.0/
URL http://hdl.handle.net/10466/15645doi: 10.1016/j.energy.2017.03.146
*1 Corresponding author. Phone: +81-72-254-9229, Fax: +81-72-254-9904,
E-mail: [email protected]
Optimal operation of heat supply systems 1
with piping network 2
3
Ryohei Yokoyama*, Hiroyuki Kitano, and Tetsuya Wakui 4
Department of Mechanical Engineering, Osaka Prefecture University 5
1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan 6
7
8 Abstract 9
It is expected that energy saving may be attained by connecting heat source 10
equipment and air conditioning equipment in multiple buildings with piping network 11
and operating heat source equipment flexibly in consideration of heat demands required 12
by air conditioning equipment. In this paper, an optimization method is proposed to 13
operate such heat supply systems with piping network rationally. Mass flow rates and 14
temperatures of water are adopted as basic variables to express heat flow rates as well as 15
pressure and heat losses in piping segments. The discreteness for the selection of 16
piping segments for water flow are also taken into account. To avoid treating the 17
nonlinearity directly, mass flow rates are discretized, and the optimization problem is 18
finally formulated as a mixed-integer linear programming one, and its suboptimal 19
solution is derived efficiently by a two-stage approach. A case study is conducted for 20
a heat supply system for space cooling and heating of an exhibition center with multiple 21
buildings. Through the study, the validity and effectiveness of the proposed 22
2
optimization method is shown in terms of solution optimality and computation time. 23
In addition, it is shown how the primary energy consumption can be reduced using 24
piping network. 25
26
Keywords: Air conditioning, Heat supply, Piping network, Optimal operation, 27
Mixed-integer linear programming 28
29
30 1. Introduction 31
In the Japanese commercial sector, the energy consumption for air conditioning 32
accounts for about one third of the total, and it is important to reduce it for energy 33
saving. In many cases, air conditioning equipment (heat exchanging units) installed in 34
each building is supplied with heat independently by heat source equipment (chilling 35
and heating units) installed in the same building. Thus, the operational flexibility is 36
very low, which leads to the operation of heat source equipment at part loads, and 37
consequently to increases in the energy consumptions not only of heat source equipment 38
but also of auxiliary equipment such as cooling towers and pumps. Therefore, it may 39
be impossible to attain energy saving only by adjusting the operational strategy. On 40
the other hand, it is expected that energy saving may be attained by connecting heat 41
source and air conditioning equipment in multiple buildings with piping network and 42
operating heat source equipment flexibly in consideration of heat demands required by 43
air conditioning equipment. However, as the operational flexibility is heightened, it may 44
become difficult to operate heat source equipment rationally. 45
3
Many optimal operational planning methods were proposed previously to operate 46
energy supply systems rationally. However, if details of the systems are taken into 47
account, optimization models are so complex that optimization problems cannot be 48
solved easily. Thus, optimization models should be so simple that optimization 49
problems can be solved easily while their results make sense for the operational 50
planning. Primarily, mass flow rates, pressures, and temperatures of water should be 51
adopted as basic variables to consider balances of mass flow rates, pressures, and heat 52
flow rates. However, optimization problems include the nonlinearity of heat flow rates 53
as well as pressure and heat losses in relation to mass flow rates and temperatures, and 54
consequently cannot be solved easily. Thus, many optimization models treat only heat 55
flow rates in place of mass flow rates, pressures, and temperatures as basic variables. 56
Heat supply systems with piping networks are typical for district heating and cooling. 57
Papers on optimization of district heating and cooling systems were published in two 58
categories. The first category is related with optimization of only heat supply systems 59
excluding piping networks. Yokoyama et al. solved an optimal operational problem 60
formulated as a mixed-integer linear programming (MILP) one by combining the 61
branch and bound method with the dynamic programming one [1]. Sakawa et al. 62
solved a similar problem using genetic algorithms [2, 3]. In these papers, only heat 63
flow rates were considered as variables whose values were to be determined by 64
optimization. The second category is related with optimization of heat supply systems 65
including piping networks. Chan et al. investigated an optimal design of distribution 66
piping networks using genetic algorithm with local search [4]. Söderman studied 67
optimization of the structure and operation of district cooling networks in urban regions 68
[5]. However, these papers treated only heat flow rates as variables whose values were 69
4
to be determined by optimization. Khir and Haouari proposed an optimization model 70
of a district cooling system in which pressure and temperature drops were taken in 71
account [6]. However, these drops were assumed to be independent of mass flow rates 72
and temperatures. In addition, in these papers, only one or few heat sources were 73
taken into account and the network structures were relatively simple, which means that 74
there were hardly alternatives for the way to supply heat to consumers. Vesterlund and 75
Dahl proposed a method of optimizing the operation of district heating systems 76
including piping loops [7, 8]. They also considered physical models for pressure and 77
temperature in relation to mass flow rates as well as multiple heat sources. However, 78
they did not treat the selection of piping segments for water flows. Guelpa et al. 79
proposed a method of optimizing the operation of large district heating networks 80
through fast simulation [9]. However, they focused only on the optimal operation of 81
pumps. In addition, they did not treat the selection of piping segments for water flows. 82
Therefore, all of these approaches cannot be applied to the aforementioned heat supply 83
system under consideration in which heat source and air conditioning equipment is 84
connected complexly with piping network. 85
In this paper, an optimization method is proposed to rationally operate a heat 86
supply system in which heat source and air conditioning equipment is connected with 87
piping network. To consider the piping network explicitly, mass flow rates and 88
temperatures of water are adopted as basic variables to express heat flow rates as well as 89
pressure and heat losses in piping segments. Thus, the optimization problem includes 90
the nonlinearity of heat flow rates as well as pressure and heat losses in relation to mass 91
flow rates and temperatures, and it cannot be necessarily convex. On the other hand, 92
the problem also includes the discreteness for the selection of piping segments used for 93
5
water flow. Therefore, the optimization problem results in a mixed-integer nonlinear 94
programming (MINLP) one, which cannot be solved easily even using commercial 95
MINLP solvers available currently. To avoid treating the aforementioned nonlinearity 96
directly, mass flow rates are discretized, and the optimization problem is converted into 97
a MILP one. Since this conversion generates many binary variables for the 98
discretization, it is still difficult to solve the problem even using commercial MILP 99
solvers available currently. Here, its suboptimal solution is derived efficiently by the 100
following two-stage approach: At the first stage, the MILP problem is solved with 101
many binary variables relaxed into continuous ones, and a lower bound for the optimal 102
value of the objective function is evaluated; At the second stage, the MILP problem is 103
solved with the mass flow rates limited, and an upper bound for the optimal value of the 104
objective function is evaluated. 105
This proposed method is applied to the optimal operation of a heat supply system 106
with sixteen pieces of heat supply equipment and nine pieces of air conditioning 107
equipment installed in six buildings for a exhibition center in Osaka, Japan. A case 108
study is conducted with different heat demands, and the validity of the solutions 109
obtained by the two-stage approach is shown based on the differences between upper 110
and lower bounds for the optimal value of the primary energy consumption as the 111
objective function to be minimized. In addition, the operation of a conventional heat 112
supply system without piping network is also investigated, and the energy saving 113
potential is clarified by comparing these two operations in terms of the primary energy 114
consumption. 115
116
117
6
2. Modeling of piping network 118
2.1. Fundamental equations 119
To model a piping network, models of piping segment and junction are shown 120
below in the case of cold water supply for space cooling. Models in the case of hot 121
water supply for space heating are obtained similarly. 122
As shown in Fig. 1, the position j (j = 1, 2, !, J(i)) , where J is the number of 123
positions, is defined for a piping segment i (i = 1, 2, !, I ) , where I is the number of 124
piping segments, and the following equations are considered for each position: 125
Relationship among mass flow rate, temperature, and heat flow rate 126 127
!Q(i, j) = c !m(i)(T !T(i, j)) (i = 1, 2, ", I ; j = 1, 2, ", J(i)) (1) 128
where !m is the mass flow rate, T is the temperature, !Q is the heat flow rate, c is the 129
specific heat of water, and T is the maximum temperature used as the reference one in 130
the target system. Since pressure energy is negligibly small as compared with internal 131
energy for water flow, enthalpy is almost equal to internal energy, and internal energy 132
flow rate is denoted by heat flow rate. 133
Heat flow rate balance and heat loss 134 135
!Q(i, j + 1) = !Q(i, j)!! !Q(i, j)
! !Q(i, j) = c !m(i)(T0(i, j)!T(i, j)) (1! e!"d(i,j)h(i,j)l(i,j)
c !m(i) )"
#
$$$$
%$$$$
(i = 1, 2, ", I ; j = 1, 2, ", J(i)!1)
(2) 136
where ! !Q is the heat flow rate for heat loss, T0 is the ambient temperature, d is the 137
inner diameter of the piping segment, l is the length of the piping segment, and h is the 138
overall heat transfer coefficient of the piping segment with insulation. This equation is 139
7
derived by integrating a differential equation for heat balance in an infinitesimal piping 140
segment. 141
Pressure loss 142 143
!P(i, j) = 10.665gl(i, j)(1 + r(i, j)) !m(i)1.85
"0.85C(i, j)1.85d(i, j)4.87 (i = 1, 2, ", I ; j = 1, 2, ", J(i)!1) 144
(3) 145
where !P is the pressure loss, g is the gravitational acceleration, r is the ratio of the 146
additional pressure loss by junctions and curvatures, ! is the mass density of water, 147
and C is the roughness coefficient of the piping segment. This equation is based on 148
the Hazen-Williams equation for the pressure loss of water flow in pipes [10]. 149
Although the pressure is defined at any position of the piping segment primarily, only 150
the pressure loss for a subsegment between any adjacent two positions is taken into 151
account. 152
On the other hand, as shown in Fig. 2, the sets for piping segments A(k) and 153
B(k) for inlet and outlet water flows, respectively, are defined at a piping junction 154
k (k = 1, 2, !, K ) , where K is the number of piping junctions, the following equations 155
are considered for each piping junction: 156
Mass flow rate balance 157 158
!m(i) =i!A(k)" !m(i)
i!B(k)" (k = 1, 2, ", K ) (4) 159
Heat flow rate balance and perfect mixing 160 161
8
!Q(i, J(i)) =i!A(k)" !Q(i, 1)
i!B(k)"
T(i, 1) =T[k ] (i ! B(k), !m(i) > 0)
#
$
%%%%
&%%%%
(k = 1, 2, ", K ) (5) 162
where the second equation means the perfect mixing, and is considered only for the 163
outlet piping segments with water flow, or !m(i) > 0 . Although the pressure balance 164
should be considered at any piping junction primarily, it is not considered here. 165
The total primary energy consumed by heat source equipment, their auxiliary 166
machinery, and pumps for circulating water between heat source and air conditioning 167
equipment is adopted as the objective function to be minimized. Here, primary energy 168
is an energy form which is not converged or transformed. The pumping power 169
consumption is considered as a part of the primary energy consumption by summing up 170
the products of pressure loss and volumetric flow rate for all the piping segments and 171
dividing it by the pump efficiency as follows: 172 173
!EPP =1!
!W(i, j)j=1
J(i)!1
"i=1
I
"
!W(i, j) =!m(i)"#P(i, j) (i = 1, 2, ", I ; j = 1, 2, ", J(i)!1)
#
$
%%%%%%
&
%%%%%%
(6) 174
where !EPP is the pumping power consumption, ! is the pump efficiency, and !W is 175
the power consumption of each the piping segment. 176
177
2.2. Linearization of nonlinear terms 178
There are some nonlinear terms in the aforementioned fundamental equations. It 179
is very difficult to directly treat the optimization model with the nonlinear terms which 180
are not necessarily convex. Here, the nonlinear terms are linearized using binary 181
9
variables, and the optimization model is converted into a mixed-integer linear one, 182
which can be treated more easily. 183
First, the mass flow rate in the piping segment !m is discretized into discrete 184
values !mn using the binary variable !n for each discrete point n (n = 1, 2, !, N(i)) , 185
where N is the number of discrete points, as follows: 186 187
!m(i) = !mn(i)!n(i) n=1
N (i)
!
!n(i)n=1
N (i)
! = 1
!n(i)" {0, 1} (n = 1, 2, ", N(i)) !m1(i) = 0
#
$
%%%%%%%%%%
&
%%%%%%%%%%
(i = 1, 2, ", I ) (7) 188
This equation means that !m(i) is selected among !m1(i) , !m2(i) , ! , !mN (i)(i) . The 189
temperature in Eq. (1) is expressed by introducing the temperature Tn corresponding 190
to the discrete value of mass flow rate !mn as follows. 191 192
T !T(i, j) = (T !Tn(i, j)n=1
N (i)
" )
T !Tn(i, j)# (T !T)!n(i) (n = 1, 2, !, N(i))
$
%
&&&&
'&&&&
(i = 1, 2, !, I ; j = 1, 2, !, J(i)) 193
(8) 194
where T is the minimum temperature used in the system. With these conversions, 195
Eq. (1) is transformed and linearized into 196 197
!Q(i, j) = c !mn(i)
n=1
N (i)
! (T "Tn(i, j)) (i = 1, 2, ", I ; j = 1, 2, ", J(i)) (9) 198
This equation together with Eqs. (7) and (8) means that !Q(i, j) is selected among 199
c !m1(i)(T !T(i, j)) , c !m2(i)(T !T(i, j)) , ! , c !mN (i)(i)(T !T(i, j)) . 200
10
Corresponding to the discretization of the mass flow rate through Eq. (7), the heat 201
flow rate for heat loss in Eq. (2) is transformed and linearized as follows: 202 203
! !Q(i, j) = q(i, j, !mn(i))n=1
N (i)
! (T0(i, j)"T )"n(i) + (T "Tn(i, j)){ }
q(i, j, !m1(i)) = 0
q(i, j, !mn(i)) = c !mn(i) (1" e"#d(i,j)h(i,j)l(i,j)
c !m n(i) ) (n = 2, 3, ", N(i))
#
$
%%%%%%%%
&
%%%%%%%% (i = 1, 2, ", I ; j = 1, 2, ", J(i)"1)
(10) 204
where q is the heat loss coefficient corresponding to the mass flow rate at each discrete 205
point. This equation together with Eqs. (7) and (8) means that ! !Q(i, j) is selected 206
among q(i, j, !m1(i))(T0(i, j)!T(i, j)) , q(i, j, !m2(i))(T0(i, j)!T(i, j)) , ! , 207
q(i, j, !mN (i)(i))(T0(i, j)!T(i, j)) . In a similar way, corresponding to the discretization 208
of the mass flow rate through Eq. (7), the pressure loss of Eq. (3) is transformed and 209
linearized as follows: 210 211
!P(i, j) = p(i, j, !mn(i))"n(i)n=1
N (i)
!
p(i, j, !mn(i)) = 10.665gl(i, j)(1 + r(i, j)) !mn(i)1.85
#0.85C(i, j)1.85d(i, j)4.87 (n = 1, 2, ", N(i))
"
#
$$$$$$
%
$$$$$$ (i = 1, 2, ", I ; j = 1, 2, ", J(i)&1)
(11) 212
where p is the pressure loss corresponding to the mass flow rate at each discrete point. 213
This equation together with Eq. (7) means that !P(i, j) is selected among 214
p(i, j, !m1(i)) , p(i, j, !m2(i)) , ! , p(i, j, !mN (i)(i)) . 215
The second equation in Eq. (5), or the perfect mixing must be considered only for 216
the outlet piping segments with water flow, or !m(i) > 0 , and is transformed into 217 218
(1! !1(i))T(i, 1) = (1! !1(i))T[k ] (k = 1, 2, !, K ; i " B(k)) (12) 219
11
where the product of the temperature T and 1! !1(i) , which has the value of 1 only 220
when !m(i) > 0 , arises. However, this nonlinear term can be linearized by replacing it 221
with continuous variables ! and ! , and adding inequality constraints as follows [11]: 222 223
!(i, k) = "(i, k)0 ! !(i, k)!T(1" #1(i))T(i, 1)"T#1(i)! !(i, k)!T(i, 1) 0 ! "(i, k)!T(1" #1(i))T[k ]"T#1(i)! "(i, k)!T[k ]
#
$
%%%%%%%
&
%%%%%%%
(k = 1, 2, !, K ; i ' B(k)) (13) 224
Finally, in a way similar to the linearization of the pressure loss of Eq. (11) 225
corresponding to the discretization of the mass flow rate through Eq. (7), the power 226
consumption of each piping segment of Eq. (6) is transformed and linearized as follows: 227 228
!W(i, j) = w(i, j, !mn(i))!n(i)n=1
N (i)
!
w(i, j, !mn(i)) = 10.665gl(i, j)(1 + r(i, j)) !mn(i)2.85
"1.85C(i, j)1.85d(i, j)4.87 (n = 1, 2, ", N(i))
"
#
$$$$$$
%
$$$$$$ (i = 1, 2, ", I ; j = 1, 2, ", J(i)&1)
(14) 229
where w is the power consumption corresponding to the mass flow rate at each discrete 230
point. This equation together with Eq. (7) means that !W(i, j) is selected among 231
w(i, j, !m1(i)) , w(i, j, !m2(i)) , ! , w(i, j, !mN (i)(i)) . 232
233
234 3. Modeling of heat source and air conditioning equipment 235
3.1. Heat source equipment 236
To complete the optimization model, it is necessary to model not only the piping 237
segments and junctions but also heat source and air conditioning equipment. Since 238
there are several types of heat source equipment, their performances cannot be modelled 239
12
generally. As an example, a model for a gas-fired absorption chilling and heating unit 240
is shown here. 241
The performance of the gas-fired absorption chilling and heating unit in the case of 242
cold water supply for space cooling is expressed by the relationship between cooling 243
output and city gas consumption using a piecewise linear equation. In addition, the 244
power consumption for auxiliary machinery such as cooling towers and pumps is also 245
formulated in relation to the city gas consumption. These relationships are expressed 246
as follows: 247 248
!QAR = (aARx fx + bARx!x )x=1
X
!
!EARa = (aARx
a fx + bARxa !x )
x=1
X
!
!FAR = fxx=1
X
!
!x " 1x=1
X
!f x!x " fx " fx!x (x = 1,2,",X) !x # {0,1} (x = 1,2,",X)
$
%
&&&&&&&&&&&&&&&&&
'
&&&&&&&&&&&&&&&&&
(15) 249
where !QAR is the cooling output, !EAR
a is the power consumption for auxiliary 250
machinery, !FAR is the city gas consumption, X is the number of divisions, x is the 251
index for divisions, ! is the binary variable for selecting a division, f is the city gas 252
consumption for a division, f and f are upper and lower limits for f, respectively, 253
and aAR , bAR , aARa , and bAR
a are performance characteristic values. 254
Generally, the performances of gas-fired absorption chilling and heating units 255
depend on the outlet cold water temperature. In considering this dependence, aAR , 256
bAR , f , and f become functions with respect to the outlet cold water temperature, 257
13
which generates another nonlinearity. To avoid this nonlinearity, the outlet cold water 258
temperature is divided into its several ranges, and Eq. (15) is considered for each range, 259
and the selection of a range is expressed by a binary variable. A detailed formulation 260
is omitted here. 261
A model in the case of hot water supply for space heating is obtained similarly. 262
263
3.2. Air conditioning equipment 264
As for air conditioning equipment, the characteristics for water and air sides should 265
be considered. However, this consideration generates nonlinear equations, and makes 266
the optimization model more complex. Thus, the characteristics only for the water 267
side is considered here. 268
A cooling or heating demand is given as a fundamental condition for each piece of 269
air conditioning equipment. In addition, the dependence of the demand on the inlet 270
water temperature and mass flow rate of water is also considered. Although this 271
relationship is generally nonlinear, and a linearized one is used here. Since the mass 272
flow rate is discretized, the relationship may not be satisfied strictly. Thus, it is 273
considered as an inequality in place of the equality. 274
275
276 4. Determination of optimal operational strategy 277
4.1. Optimization problem 278
In the optimization problem, the equations for piping segments and junctions as 279
well as heat source and air conditioning equipment are considered as constraints to be 280
satisfied. The cooling or heating outputs of the heat source equipment are added to the 281
14
heat flow rate balance equations of the corresponding piping segments. On the other 282
hand, the cooling or heating demands of the air conditioning equipment are subtracted 283
from the heat flow rate balance equations of the corresponding piping segments. 284
Although pressure losses of heat source and air conditioning equipment are not 285
described above, they are evaluated in relation to the mass flow rates, and are added to 286
the pressure losses of the corresponding piping segments. 287
As aforementioned, the objective function to be minimized by the optimization is 288
the total primary energy consumption, which is composed of those for heat source 289
equipment, their auxiliary machinery, and pumps for circulating water between heat 290
source and air conditioning equipment. 291
292
4.2. Solution method 293
As aforementioned, mass flow rates are discretized to avoid the nonlinearity, and 294
the optimization problem is converted into an MILP one. Although this conversion 295
still keeps the exactness of the problem for discretized values of mass flow rates, it 296
generates many binary variables for the discretization. Thus, it is still difficult to solve 297
the problem even using commercial MILP solvers available currently. For the purpose 298
of applying the proposed method to real-time operation of the systems, it is necessary to 299
obtain a solution in a reasonable computation time. Here, its suboptimal solution is 300
derived efficiently by the following two-stage approach: At the first stage, the MILP 301
problem is solved with the binary variables for zero mass flow rates !1(i) taken into 302
account directly and the binary variables for nonzero mass flow rates 303
!n(i) (n = 2, 3, !, N(i)) relaxed into continuous ones, and a lower bound for the 304
optimal value of the objective function is evaluated; At the second stage, the MILP 305
15
problem is solved with the mass flow rates limited based on the solution obtained at the 306
first stage, and an upper bound for the optimal value of the objective function is 307
evaluated; If the upper and lower bounds differ slightly, the solution obtained at the 308
second stage can be considered as a good suboptimal one close to the optimal one. 309
310
4.3. Visualization of modeling and optimization results 311
To conduct the optimization calculation, it is necessary to define variables, 312
constraints, and objective function. For the optimization problem under consideration, 313
there are many variables and constraints for piping segments and junctions, and the 314
optimization model is very complex. Thus, it is difficult to define them in a manual 315
way. To conduct the modeling efficiently, only a minimum number of data which 316
defines piping segments and junctions is input manually. To avoid mistakes in the 317
modelling, input data can be checked easily by visualizing information on piping 318
segments and junctions. In addition, the optimization model which includes the 319
definition of variables, constraints, and objective function can be generated 320
automatically using the input data. The results obtained by the optimization 321
calculation include many data. Among them, the mass flow rates, temperatures, and 322
heat flow rates for heat source and air conditioning equipment are displayed 323
automatically. The mass flow rates and temperatures for piping segments are 324
displayed by the thickness and color of the lines, respectively, so that they can be 325
understood easily. 326
327
328 5. Case study 329
16
5.1. Target system 330
A heat supply system which supplies cold and hot water for space cooling and 331
heating, respectively, to an exhibition center with multiple buildings in Osaka, Japan is 332
investigated in this case study. Figure 3 shows the configuration of this target system. 333
The system is composed of fifteen pieces of heat source equipment (R1~R15) and nine 334
air conditioning units (AC1~AC9), which are expressed by rectangles, as well as many 335
piping segments, which are expressed by lines. The heat source equipment is 336
composed of ten gas-fired absorption chilling and heating units (R1~R3, R5~R9, R14, 337
and R15), one centrifugal chilling unit (R4), and four heat pump chilling and heating 338
units (R10~R13). The gas-fired absorption and heat pump chilling and heating units 339
have already existed, while the centrifugal chilling unit has been installed newly. Here, 340
pumps are not directly taken into account, and the power consumptions of pumps for 341
circulating water between heat source and air conditioning equipment are indirectly 342
taken into account based on the pressure losses and mass flow rates in the piping 343
segments. Solid lines denote piping segments for water supply and return flows, while 344
broken lines denote piping segments for water bypass flows. Each piping segment is 345
identified by one or two red numbers. The directions of water flows are determined 346
and undetermined for the piping segments with and without arrows, respectively. 347
Valves are not directly taken into account, and their switches are expressed by 348
determining water flows. Each piping junction is identified by a blue number. Only 349
the cold water supply for space cooling is considered here. 350
351
5.2. Conditions 352
The cooling capacities or the cooling outputs at the rated load of the gas-fired 353
17
absorption chilling and heating units, centrifugal chilling unit, and heat pump chilling 354
and heating units are shown in Table 1. As an example, Fig. 4 shows the relationship 355
between the cooling output as well as the power consumption for auxiliary machinery 356
and the city gas consumption at part load for the gas-fired absorption chilling and 357
heating unit R3. Since the gas-fired absorption and heat pump chilling and heating 358
units have been operated for a long period, their performance degradations are estimated, 359
and the current performances are evaluated based on the original performances and 360
performance degradations. Table 2 shows cooling demands given to the air 361
conditioning equipment. As shown in this table, six cases I to VI for the cooling 362
demands are set, and an upper limit for the inlet water temperature for each air 363
conditioning unit is given. 364
The discretization width of the mass flow rate is set at 2.0 kg/s, and the minimum 365
and maximum temperatures are set at 7.0 and 20.0 °C, respectively. The following 366
additional constraints on mass flow rates are considered: When water flow is 367
restricted by switching valves, a corresponding constraint is added; When cooling 368
demand is zero for an air conditioning unit, a corresponding constraint is added; When 369
a chilling and heating unit or a chilling unit is not operated, a corresponding constraint 370
is added. It is assumed that primary pumps for the gas-fired absorption and heat pump 371
chilling and heating units are operated with constant mass flow rates, and that a primary 372
pump for the centrifugal chilling unit is operated with variable mass flow rate. 373
The total primary energy consumption as the objective function to be minimized is 374
defined as follows: 375 376
z = !gas !FAR
AR! +!elec ( !ETR
TR! + !EAR
a
AR! + !ETR
a
TR! + !EPP ) (16) 377
18
where !ETR and !ETRa are the power consumptions for a centrifugal chilling unit or a 378
heat pump chilling and heating unit and its auxiliary machinery, respectively, AR! 379
and TR! denote the summation for all the gas-fired absorption chilling and heating 380
units and all the centrifugal chilling unit and heat pump chilling and heating units, 381
respectively, and !gas and !elec are the coefficients for primary energy consumption 382
of city gas and power, respectively. 383
To investigate the effect of the piping network which connects all the buildings, 384
the optimization calculation is conducted for the following two energy supply systems, 385
and their performances are analyzed and compared with each other: One is the system 386
shown in Fig. 3 (system A), and the other is a conventional energy supply system which 387
supplies cold water for space cooling to each building by a gas-fired absorption or heat 388
pump chilling and heating unit independently (system B). The performance of the 389
centrifugal chilling unit is much higher than those of the gas-fired absorption and heat 390
pump chilling and heating units, the centrifugal chilling unit is excluded from both the 391
systems. In addition, it is assumed that the gas-fired absorption chilling and heating 392
units R7 and R15 are forced to be stopped. 393
A HP Z840 Workstation with Intel XEON E5-2687W (8 cores, 3.1 GHz, 64 GB) is 394
used for all the optimization calculations. CPLEX Ver 12.6.1.0 is used as a 395
commercial MILP solver through a modeling system GAMS Ver 12.4.1 [12]. 396
397
5.3. Results and discussion 398
First, the optimization calculation for system A is conducted in the six cases I to VI 399
for the cooling demands set as conditions, and the results are shown in Table 3. The 400
total and contents of the primary energy consumption as the objective function to be 401
19
minimized as well as the computation time are shown at the first and second stages of 402
the two-stage approach to the solution method. The differences in the value of the 403
objective function between the first and second stages are less than 0.2 % of that 404
obtained at the second stage. This means that the two-stage approach can rationally 405
obtain a good suboptimal solution close to the optimal one. In addition, the calculation 406
time is less than 60 s except in case II, and the optimization calculation can be 407
conducted very efficiently. On the other hand, it takes 1324.38 s to conduct the 408
optimization calculation by the conventional approach in case II. The value of the 409
objective function for the optimal solution obtained by the conventional approach 410
coincides with that for the suboptimal solution obtained at the second stage by the 411
two-stage approach. Thus, the two-stage approach is very effective in terms of 412
solution optimality and computation time. 413
Next, the optimization calculation for system B is conducted in the six cases I to 414
VI for the cooling demands set as conditions, and the results are shown in Table 4. 415
Similarly, the total and contents of the primary energy consumption as the objective 416
function to be minimized as well as computation time are shown at the first and second 417
stages of the two-stage approach to the solution method. The upper and lower bounds 418
for the optimal value of the objective function coincide with each other, and the 419
suboptimal solution obtained at the second stage can be judged to be optimal. The 420
computation time is only less than 1 s. These are because the number of alternatives 421
for the operational strategy is small. 422
In comparing the total primary energy consumptions for systems A and B, it can be 423
reduced by 10.2 to 25.2 % using the piping network. In comparing the contents of the 424
primary energy consumptions for systems A and B, both the primal energy 425
20
consumptions of heat source equipment and their auxiliary machinery are reduced using 426
the piping network. This is because the energy saving can be attained by operating 427
heat source equipment with higher performances concentrically. The primary energy 428
consumption of pumps is also reduced using the piping network. Generally, it is 429
necessary to circulate cold water for longer distances using the piping network, and it is 430
afraid that the pumping power consumption increases. Cold water with large mass 431
flow rates flows through heat source and air conditioning equipment by pumps with 432
constant mass flow rates in system B. On the other hand, cold water flows from heat 433
source equipment to air conditioning equipment are dispersed, and the mass flow rates 434
of cold water which flows through air conditioning equipment can be decreased. Thus, 435
the pumping power consumption for system A can be smaller than that for system B. 436
As an example, the heat supply patterns for systems A and B in case III are shown 437
in Figs. 5 and 6, respectively. According to the heat supply pattern for system A in Fig. 438
5, although the buildings with cooling demands are dispersed, the cooling demands not 439
only in building 6 but also in buildings 2 and 5 are supplied by the gas-fired absorption 440
chilling and heating units installed in building 6. However, the cooling demands in 441
building 4 are supplied independently by the heat pump chilling and heating units 442
installed in the same building. 443
As an example, the cooling outputs of heat source equipment for systems A and B 444
in case III are shown in Table 5. The total cooling output of the gas-fired absorption 445
chilling and heating units for system A are slightly larger than that for system B because 446
of an increase in the heat loss from the piping. 447
448
449
21
6. Conclusions 450
In this paper, an optimization method has been proposed to operate heat supply 451
systems in which heat source and air conditioning equipment in multiple buildings are 452
connected with piping network. To avoid the nonlinearity of heat flow rates as well as 453
pressure and heat losses in relation to mass flow rates and temperatures, mass flow rates 454
have been discretized, and the optimization problem has been converted into a MILP 455
one. In addition, its suboptimal solution has been derived efficiently by the two-stage 456
approach. The proposed method is applied to the optimal operation of an actual heat 457
supply system which supplies cold and hot water for space cooling and heating, 458
respectively, to an exhibition center with multiple buildings. The following results 459
have been obtained through the case study in the case of cold water supply for space 460
cooling: 461
• The differences in the value of the objective function between the first and second 462
stages are small enough as compared with that obtained at the second stage. This 463
means that the optimization calculation based on the two-stage approach can obtain a 464
good suboptimal solution close to the optimal one rationally. 465
• The computation time by the two-stage approach is much shorter than that by the 466
conventional approach. This means that the two-stage approach can derive a 467
suboptimal solution very efficiently. 468
• The primary energy consumption can be reduced by 10.2 to 25.2 % for different 469
cooling demands using piping network. All the primary energy consumptions of heat 470
source equipment, auxiliary machinery, and pumps can be reduced. 471
• Even when buildings with cooling demands are dispersed, all the cooling demands 472
are supplied by operating heat source equipment with higher performances 473
22
concentrically and using piping network. 474
As subsequent subjects, it is important to grasp the performances of heat source 475
equipment by measurement and apply the proposed method to actual operation of the 476
target heat supply system. It is also important to extend the proposed method so that it 477
can be applied to the optimal operation for consecutive periods as well as the optimal 478
design of heat supply systems with piping network. 479
480
481 Acknowledgments 482
A part of this work has been done in relation to the project “Regional Energy 483
Network Gathering Existing Energy Resources Through Rail Network,” in the Low 484
Carbon Technology Research and Development Program, Ministry of the Environment, 485
Japan. 486
487
488 Nomenclature 489
A : set for inlet piping segments at piping junction 490
a : coefficient for performance characteristics of heat source equipment, kW/(m2/s) 491
B : set for outlet piping segments at piping junction 492
b : coefficient for performance characteristics of heat source equipment, kW 493
C : roughness coefficient of piping segment 494
c : specific heat of water, kJ/(kg·°C) 495
d : inner diameter of piping segment, m 496
!E : electric power consumption of heat source equipment or pumps, kW 497
23
!F : city gas consumption of heat source equipment, m3/s 498
f : city gas consumption for division, m3/s 499
g : gravitational acceleration, m/s2 500
h : overall heat transfer coefficient of piping segment with insulation, kW/(m2·°C) 501
I : number of piping segments 502
J : number of positions 503
K : number of piping junctions 504
l : length of piping segment, m 505
!m : mass flow rate, kg/s 506
N : number of discrete points 507
!P : pressure loss of piping segment, Pa 508
p : pressure loss at discrete point, kW 509
!Q : heat flow rate, kW 510
! !Q : heat flow rate for heat loss, kW 511
q : heat loss coefficient at discrete point, kW 512
r : ratio of additional pressure loss by piping junctions and curvatures 513
T : temperature, °C 514
T0 : ambient temperature, °C 515
!W : power consumption of piping segment, kW 516
w : power consumption at discrete point of piping segment, kW 517
X : number of divisions 518
( )
: upper limit 519
( )
: lower limit 520
521
24
Greek symbols 522
! : selection of division 523
! : selection of discrete point 524
! : product of T[k] and 1! !1(i) , °C 525
! : pump efficiency 526
! : product of T(i, 1) and 1! !1(i) , °C 527
! : mass density of water, kg/m3 528
AR! : summation for all gas-fired absorption chilling and heating units 529
TR! : summation for all centrifugal chilling unit and heat pump chilling and heating 530
units 531
! : coefficient for primary energy consumption 532
533
Subscripts and superscripts 534
AR : gas-fired absorption chilling and heating unit 535
a : auxiliary machinery 536
elec : power consumption 537
gas : city gas consumption 538
n : index for discrete points 539
PP : pumps 540
TR : centrifugal chilling unit or heat pump chilling and heating unit 541
x : index for divisions 542
543
Arguments 544
i : index for piping segments 545
25
j : index for positions 546
k : index for piping junctions 547
548
549 References 550
[1] Yokoyama R, Ito K, Kamimura K, Miyasaka F. Development and evaluation of an 551
advisory system for optimal operation of a district heating and cooling plant. In: 552
Proceedings of the International Conference on Renewable and Advanced Energy 553
Systems for the 21st Century. Paper No. RAES99-7641: 1–9; 1999. 554
[2] Sakawa M, Kato K, Ushiro S, Inaoka M. Operation planning of district heating and 555
cooling plants using genetic algorithms for mixed integer programming. Applied 556
Soft Computing 2001; 1 (2): 139–150. 557
[3] Sakawa M, Kato K, Ushiro S. Operational planning of district heating and cooling 558
plants through genetic algorithms for mixed 0-1 linear programming. European 559
Journal of Operational Research 2002; 137 (3): 677–787. 560
[4] Chan ALS, Hanby VI, Chow TT. Optimization of distribution piping network in 561
district cooling system using genetic algorithm with local search. Energy 562
Conversion and Management 2007; 48 (10): 2622–2629. 563
[5] Söderman J. Optimization of structure and operation of district cooling networks in 564
urban regions. Applied Thermal Energy 2007; 27 (16): 2665–2676. 565
[6] Khir R, Haouari M. Optimization models for a single-plant district cooling system. 566
European Journal of Operational Research 2015; 247 (2): 648–658. 567
[7] Vesterlund M, Dahl J. A method for the simulation and optimization of district 568
heating systems with meshed networks. Energy Conversion and Management 569
26
2015; 89: 555–567. 570
[8] Vesterlund M, Toffolo A, Dahl J. Optimization of multi-source complex district 571
heating network, a case study. In: Proceedings of the 29th International Conference 572
on Efficiency, Cost, Optimization, Simulation and Environmental Impact of 573
Energy Systems (ECOS 2016). Paper No. 299: 1–11; 2016. 574
[9] Guelpa E, Toro C, Sciacovelli A, Melli R, Sciubba E, Verda V. Optimal operation 575
of large district heating networks through fast fluid-dynamic simulation. Energy 576
2016; 102: 586–595. 577
[10] Hazen A, Williams GS. Hydraulic tables, 3rd ed. New York: John Wiley and Sons; 578
1920. 579
[11] Glover F. Improved linear integer programming formulations of nonlinear integer 580
problems. Management Science 1975; 22 (4): 455–460. 581
[12] Rosenthal RE. GAMS—a user’s guide. Washington, DC: GAMS Development 582
Corp; 2012. 583
584
27
Captions for tables and figures 585
Table 1 Cooling capacities of heat source equipment 586
Table 2 Cooling demands 587
Table 3 Values of objective function and computation times for system A 588
Table 4 Values of objective function and computation times for system B 589
Table 5 Cooling outputs of heat source equipment in case III 590
Fig. 1 Definition of variables for piping segment 591
Fig. 2 Definition of variables for piping junction 592
Fig. 3 Configuration of heat supply system with piping network 593
Fig. 4 Performance of gas-fired absorption chilling and heating unit R3 594
Fig. 5 Optimal heat supply for system A in case III 595
Fig. 6 Optimal heat supply for system B in case III 596
597
28
Table 1 Cooling capacities of heat source equipment 598
599
600
Heat source equipment
Cooling capacity
MW (RT)
Heat source equipment
Cooling capacity
MW (RT) R1 2.814 (800) R9 1.056 (300) R2 2.110 (600) R10 0.358 (102) R3 2.110 (600) R11 0.358 (102) R4 1.759 (500) R12 0.358 (102) R5 4.398 (1250) R13 0.358 (102) R6 4.398 (1250) R14 1.933 (550) R7 3.517 (1000) R15 0.985 (280) R8 3.517 (1000) Total 30.029 (8538)
601
602
29
Table 2 Cooling demands 603
604
605 (Unit: MW) 606
Case AC1 AC2 AC3 AC4 AC5 AC6 AC7 AC8 AC9 Total
I 1.0 — 0.6 — — — — — 0.8 2.4 II 0.8 0.9 — 1.7 0.2 — 0.2 — — 3.8 III — 1.5 — 3.0 — 0.2 0.2 — 1.0 5.9 IV 1.0* — 1.0 — — — — — — 2.0 V 1.0 — 1.0* — — — — — — 2.0 VI 1.2* — 1.2 — 0.2* 0.2* — — — 2.8
Value with *: upper limit for inlet water temperature of 12 °C. 607 Value without *: upper limit for inlet water temperature of 8 °C. 608
609 610
30
Table 3 Values of objective function and computation times for system A 611
612
613
Case Stage Primary energy consumption MW Compu-
tation time s
Heat source equipment
Auxiliary equipment Pump Total
I 1st 3.348 0.641 0.165 4.154 27.22 2nd 3.348 0.641 0.173 4.162 11.99
II 1st 5.313 1.201 0.248 6.762 200.2 2nd 5.313 1.201 0.251 6.765 306.1
III 1st 8.246 1.617 0.486 10.35 22.70 2nd 8.246 1.617 0.488 10.35 16.72
IV 1st 2.788 0.619 0.159 3.566 22.56 2nd 2.787 0.619 0.165 3.571 8.02
V 1st 2.788 0.619 0.159 3.566 20.32 2nd 2.787 0.619 0.165 3.571 12.69
VI 1st 3.924 0.641 0.265 4.830 50.11 2nd 3.925 0.641 0.271 4.837 3.36
614 615
616
31
Table 4 Values of objective function and computation times for system B 617
618
619
Case Stage Primary energy consumption MW Compu-
tation time s
Heat source equipment
Auxiliary equipment Pump Total
I 1st 4.070 1.155 0.342 5.568 0.29 2nd 4.070 1.155 0.342 5.568 0.14
II 1st 6.125 1.339 0.533 7.997 0.38 2nd 6.125 1.339 0.533 7.997 0.18
III 1st 9.293 1.650 0.591 11.53 0.26 2nd 9.293 1.650 0.591 11.53 0.14
IV 1st 3.412 0.864 0.274 4.550 0.29 2nd 3.412 0.864 0.274 4.550 0.14
V 1st 3.413 0.864 0.274 4.551 0.27 2nd 3.413 0.864 0.274 4.551 0.14
VI 1st 4.670 0.891 0.319 5.880 0.32 2nd 4.670 0.891 0.319 5.880 0.15
620 621
32
Table 5 Cooling outputs of heat source equipment in case III 622
623
624 Heat source equipment
Cooling output MW Heat source equipment
Cooling output MW System A System B System A System B
R1 — 1.502 R9 — 0.561 R2 — — R10 — — R3 — — R11 0.200 0.200 R4 — — R12 0.200 0.200 R5 — — R13 — — R6 3.072 — R14 — 1.002 R7 — — R15 — — R8 2.461 2.461 Total 5.933 5.926
625 626
33
627
628
Fig. 1 Definition of variables for piping segment 629 630
• • • •
···
··· !m(i)
T(i,1)!Q(i,1)
T(i,2)!Q(i,2)
T(i,J(i))!Q(i,J(i))
T(i,J(i)‒1)!Q(i,J(i)‒1)
34
631
632
Fig. 2 Definition of variables for piping junction 633 634
!m(i), T(i,J(i)), !Q(i,J(i))(i ∈ A(k))
··· ···
!m(i), T(i,1), !Q(i,1)(i ∈ B(k))
T[k ]
35
635
636
Fig. 3 Configuration of heat supply system with piping network 637
AC5
R10
R11
R12
R13
R14
AC6
AC9
AC8
AC7
135
136
111
112
104
105
9495
61
6263
101
102
91
92
121
131
133
124
122
206
306
207
307
308
208
209
309
210
310
211
311
122
121
102
9192
6113
1
411
412
413
414
511
512
513
514
412
413
414207
208
209
210
211
212
R15
114
115
125
101
111
112
132
415
416
417
418
419
420
421
422
515
516
517
518
519
520
521
522
423
523
139
138
137
415
416
417
418
419
420
421
422
423
132
143
144
141
142
R4
427
426
425
424
424
524
427
527
426
526
425
525
6566
6496
106
116
126
62
1.75
9MW
0.35
8MW
0.35
8MW
0.35
8MW
0.35
8MW
1.93
3MW
0.98
5MW
1005
1006
134
103
9312
311
3
AC1
R1
R2
AC2
R3
AC3
2122 23
2425
3132
33
41
5152
53
747372 77
8271
203
303
204
205
305
212
312
11
31
72
401
402
403
404
405
406
407
408
409
410
501
502
503
504
505
506
507
508
509
510
401
402
403
404
405
406
407
408
409
410
411
201
202
204
205
206
1112
1375 76
7881
79
2122
51
71
201
301
202
302
304
2726
203
8328
36 3734 35
84 8586 87
428
428
528
3839
4054
5556
AC4
R5
R6
R7
R8
R9
3252
2.81
4MW
2.11
0MW
2.11
0MW
4.39
8MW
4.39
8MW
3.51
7MW
3.51
7MW
1.05
6MW
1001
1002
1003
1004
14
Bldg
. 1Bl
dg. 3
Bldg
. 2
Bldg
. 6
Bldg
. 4Bl
dg. 4
Bldg
. 4Bl
dg. 4
Bldg
. 5
36
638
639
Fig. 4 Performance of gas-fired absorption chilling and heating unit R3 640 641
2.5
2.0
1.5
1.0
0.5
0.0200150100500Co
oling
out
put
MW
Auxil
iary p
ower
cons
umpt
ion
MW
City gas consumption m3/ h
Cooling output
Auxiliary power consumption
37
642 643
Fig. 5 Optimal heat supply for system A in case III 644 645
38
646 647
Fig. 6 Optimal heat supply for system B in case III 648 649