3. Mathematical model development

#### 3.1. Physical model

To obtain the mathematical model of the DHS, it is required to design the heating system illustrated in Figure 1. The design parameters are given in Table 1. The DHS is designed based on these parameters, which could be utilized to develop mathematical model and simulations.

#### 3.2. Assumption of model development

The designed DHSs are a very complex system from mathematic modeling point of view because of the multiple connections among the subsystems. To simplify the dynamic model development process, several assumptions are listed below without affecting major properties of the DHSs [13]:



In Eq. (1), the net heat stored in the water of the boiler body is computed with the difference between the heat from the gas combustion and the heat transferred to the circulation water in the primary system. Note that the boiler efficiency is calculated according to measured opera-

Advanced Control Strategies with Simulations for a Typical District Heating System to Approaching Energy…

The return water temperatures from each substation in the primary system are given in Eqs. (2)–(4). The net heat stored in the heat exchanger (primary side) is computed between the heat from the pipe network to the substation and the heat transferred in the substation:

The supply water temperature from the substation in the secondary system is presented in Eqs. (5)–(7). The net heat stored in the heat exchanger in the secondary side is related to the heat transferred in the substation and the heat taken from the substation to the secondary system. Note that the logarithmic mean temperature difference (LMTD) is calculated in Eq. (8). Note that it refers to each substation from 1 to 3. Letter i denotes to 1–3, which is the number of substation:

LMTDi <sup>¼</sup> ½ � ð Þ� Ts1in � Ts2zi ð Þ Tr1<sup>i</sup> � Tr2im ln Ts1in � Ts2zi

dt <sup>¼</sup> cwð Þ <sup>u</sup>21G21<sup>d</sup> � <sup>0</sup>:5Gmk<sup>21</sup> ð Þ� Ts21<sup>z</sup> � Tr<sup>21</sup> <sup>f</sup> ht1Uht1½ � <sup>0</sup>:5ð Þ� Ts21<sup>z</sup> <sup>þ</sup> Tr<sup>21</sup> Tz<sup>1</sup> ð Þ <sup>1</sup>þk<sup>1</sup> (9)

dt <sup>¼</sup> cwð Þ <sup>u</sup>22G22<sup>d</sup> � <sup>0</sup>:5Gmk<sup>22</sup> ð Þ� Ts22<sup>z</sup> � Tr<sup>22</sup> <sup>f</sup> ht2Uht2½ � <sup>0</sup>:5ð Þ� Ts22<sup>z</sup> <sup>þ</sup> Tr<sup>22</sup> Tz<sup>2</sup> ð Þ <sup>1</sup>þk<sup>2</sup> (10)

dt <sup>¼</sup> cwð Þ <sup>u</sup>23G23<sup>d</sup> � <sup>0</sup>:5Gmk<sup>23</sup> ð Þ� Ts23<sup>z</sup> � Tr<sup>23</sup> <sup>f</sup> rf Urf ½ � <sup>0</sup>:5ð Þ� Ts23<sup>z</sup> <sup>þ</sup> Tr<sup>23</sup> Tz<sup>3</sup> ð Þ <sup>1</sup>þk<sup>3</sup> (11)

dt <sup>¼</sup> cwð Þ <sup>u</sup>11G11<sup>d</sup> � <sup>0</sup>:5Gmk<sup>11</sup> ð Þ� Ts1B<sup>1</sup> � Tr<sup>11</sup> <sup>f</sup> ex1Uex1LMTD1 (2)

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dt <sup>¼</sup> cwð Þ <sup>u</sup>12G11<sup>d</sup> � <sup>0</sup>:5Gmk<sup>12</sup> ð Þ� Ts1c<sup>1</sup> � Tr<sup>12</sup> <sup>f</sup> ex2Uex2LMTD2 (3)

dt <sup>¼</sup> cwð Þ <sup>u</sup>13G13<sup>d</sup> � <sup>0</sup>:5Gmk<sup>13</sup> ð Þ� Ts1<sup>D</sup> � Tr<sup>13</sup> <sup>f</sup> ex3Uex3LMTD3 (4)

dt <sup>¼</sup> <sup>f</sup> ex1Uex1LMTD1 � cwu21G21dð Þ Ts<sup>21</sup> � Tr21<sup>m</sup> (5)

dt <sup>¼</sup> <sup>f</sup> ex2Uex2LMTD2 � cwu22G22dð Þ Ts<sup>22</sup> � Tr22<sup>m</sup> (6)

dt <sup>¼</sup> <sup>f</sup> ex3Uex3LMTD3 � cwu23G23dð Þ Ts<sup>23</sup> � Tr23<sup>m</sup> (7)

Tr1<sup>i</sup> � Tr2im �<sup>1</sup>

(8)

tional data.

3.3.2. Substation model

3.3.3. Radiator model

dTr21

dTr22

Cht<sup>1</sup>

Cht<sup>2</sup>

Crf dTr23 Cex<sup>11</sup>

Cex<sup>12</sup>

Cex<sup>13</sup>

dTr11

dTr12

dTr13

Cex<sup>21</sup>

Cex<sup>22</sup>

Cex<sup>23</sup>

dTs21

dTs22

dTs23

Table 1. Design parameters of the DHS.

#### 3.3. Dynamic modeling

By applying for the first law of thermodynamics and mass conservation principle, each subsystem dynamic model is shown and described briefly below:

#### 3.3.1. Boiler model

$$\mathbf{C}\_{b}\frac{dT\_{s1}}{dt} = \mathbf{u}\_{f}\mathbf{G}\_{f\mathbf{d}}HV\eta\_{b} - \mathbf{c}\_{w}(\mathbf{u}\_{11}\mathbf{G}\_{11d} + \mathbf{u}\_{12}\mathbf{G}\_{12d} + \mathbf{u}\_{13}\mathbf{G}\_{13d})(T\_{s1} - T\_{r1\mathbf{m}})\tag{1}$$

In Eq. (1), the net heat stored in the water of the boiler body is computed with the difference between the heat from the gas combustion and the heat transferred to the circulation water in the primary system. Note that the boiler efficiency is calculated according to measured operational data.

#### 3.3.2. Substation model

$$\mathcal{L}\_{\rm ex1} \frac{dT\_{r11}}{dt} = \mathcal{c}\_{w} (\mu\_{11} \mathcal{G}\_{11d} - 0.5 \mathcal{G}\_{\rm wk11}) (T\_{s1 \to 1} - T\_{r11}) - f\_{\rm ex1} l L\_{\rm ex1} L M T D\_{1} \tag{2}$$

$$\mathcal{L}\_{ex12}\frac{dT\_{r12}}{dt} = \mathcal{c}\_w(u\_{12}\mathcal{G}\_{11d} - 0.5\mathcal{G}\_{mk12})(T\_{s1c1} - T\_{r12}) - f\_{ex2}\mathcal{U}\_{cx2}\text{LMTD}\_2\tag{3}$$

$$\mathcal{L}\_{\rm ex13} \frac{dT\_{r13}}{dt} = c\_w (u\_{13} \mathcal{G}\_{13d} - 0.5 \mathcal{G}\_{mk13}) (T\_{s1D} - T\_{r13}) - f\_{\rm ex3} \mathcal{U}\_{\rm ex3} LMTD\_3 \tag{4}$$

The return water temperatures from each substation in the primary system are given in Eqs. (2)–(4). The net heat stored in the heat exchanger (primary side) is computed between the heat from the pipe network to the substation and the heat transferred in the substation:

$$\mathcal{C}\_{\text{ex21}} \frac{dT\_{s21}}{dt} = f\_{\text{ex1}} \mathcal{U}\_{\text{ex1}} \text{LMTD}\_1 - c\_w \mu\_{21} \mathcal{G}\_{21d} (T\_{s21} - T\_{r21m}) \tag{5}$$

$$\mathcal{L}\_{\text{ex22}} \frac{dT\_{s22}}{dt} = f\_{\text{ex2}} \mathcal{U}\_{\text{ex2}} \text{LMTD}\_2 - c\_w u\_{22} \mathcal{G}\_{22d} (T\_{s22} - T\_{r22m}) \tag{6}$$

$$\mathcal{L}\_{\text{ex23}} \frac{dT\_{s23}}{dt} = f\_{\text{ex3}} \mathcal{U}\_{\text{ex3}} \text{LMTD}\_3 - c\_w u\_{23} \mathcal{G}\_{23d} (T\_{s23} - T\_{r23w}) \tag{7}$$

The supply water temperature from the substation in the secondary system is presented in Eqs. (5)–(7). The net heat stored in the heat exchanger in the secondary side is related to the heat transferred in the substation and the heat taken from the substation to the secondary system. Note that the logarithmic mean temperature difference (LMTD) is calculated in Eq. (8). Note that it refers to each substation from 1 to 3. Letter i denotes to 1–3, which is the number of substation:

$$LMTD\_i = \left[ (T\_{s1in} - T\_{s2zi}) - (T\_{r1i} - T\_{r2im}) \right] \left[ \ln \left( \frac{T\_{s1in} - T\_{s2zi}}{T\_{r1i} - T\_{r2im}} \right) \right]^{-1} \tag{8}$$

3.3.3. Radiator model

3.3. Dynamic modeling

Table 1. Design parameters of the DHS.

Cb dTs1

3.3.1. Boiler model

By applying for the first law of thermodynamics and mass conservation principle, each

No. Name Unit Data Remark

7 Supply water temperature in the secondary system �C 75 Radiator 8 Return water temperature in the secondary system �C 50 Radiator 9 Supply water temperature in the secondary system �C 50 Floor heating 10 Return water temperature in the secondary system �C 40 Floor heating 11 Indoor air temperature �C 20 Radiator terminal 12 Indoor air temperature �C 18 Floor heating terminal

13 Heated floor area in Substation #1 m2 50,000 14 Heating load index in Substation #1 W/m2 55 15 Water volume in the radiator of Substation #1 T 75 16 Factor of the heat transfer coefficient test in Substation #1 0.35 17 Heated floor area in Substation #2 m2 35,000 18 Heating load index in Substation #2 W/m2 42 19 Water volume in the radiator of Substation #2 T 38 20 Factor of the heat transfer coefficient test in Substation #2 0.28 21 Heated floor area in Substation #3 m2 40,000 22 Heating load index in Substation #3 W/m2 35 23 Water volume in the radiator of Substation #3 T 38 24 Factor related to heat transfer coefficient simulation in Substation #3 0.04 25 Heating load in Substation #1 MW 2.75 26 Heating load in Substation #2 MW 1.47 27 Heating load in Substation #3 MW 1.40

1 Outside air temperature �C �20 2 Supply water temperature in the primary system �C 120 3 Return water temperature in the primary system �C 60 4 Heat capacity in the heat source MW 7 5 Natural gas-fired boiler % 92 6 Water volume in the boiler body T 3

96 Sustainable Buildings - Interaction Between a Holistic Conceptual Act and Materials Properties

dt <sup>¼</sup> uf GfdHVη<sup>b</sup> � cwð Þ <sup>u</sup>11G11<sup>d</sup> <sup>þ</sup> <sup>u</sup>12G12<sup>d</sup> <sup>þ</sup> <sup>u</sup>13G13<sup>d</sup> ð Þ Ts<sup>1</sup> � Tr1<sup>m</sup> (1)

subsystem dynamic model is shown and described briefly below:

$$\mathbf{C}\_{\mathrm{ht1}}\frac{dT\_{r21}}{dt} = c\_w(\mu\_{21}\mathbf{G}\_{21d} - 0.5\mathbf{G}\_{\mathrm{ml21}})(T\_{s21z} - T\_{r21}) - f\_{\mathrm{ht1}}\mathbf{U}\_{\mathrm{ht1}}[0.5(T\_{s21z} + T\_{r21}) - T\_{z1}]^{(1+\mathrm{kl})} \tag{9}$$

$$\mathbf{C}\_{\text{h2}} \frac{dT\_{r22}}{dt} = \mathbf{c}\_w (\mathbf{u}\_{22} \mathbf{G}\_{22d} - \mathbf{0}.5 \mathbf{G}\_{\text{m22}}) (T\_{s22z} - T\_{r22}) - f\_{\text{h2}} \mathbf{U}\_{\text{h2}} [\mathbf{0}.5 (T\_{s22z} + T\_{r22}) - T\_{r2}]^{(1+\mathbf{k}2)} \tag{10}$$

$$\mathbf{C}\_{\eta'} \frac{dT\_{r23}}{dt} = c\_w (\mu\_{23} \mathbf{G}\_{23d} - 0.5 \mathbf{G}\_{ml23}) (T\_{s23z} - T\_{r23}) - f\_{\eta'} \mathbf{U}\_{\eta'} [0.5(T\_{s23z} + T\_{r23}) - T\_{z3}]^{(1+k3)} \tag{11}$$

The return water temperature from the end-user (radiator and radiant floor heating) is addressed in Eqs. (9)–(11). The net heat stored in the terminal equals to the heat difference between the heat gathered from the circulation water and emitted to the indoor air.

4. Actual dynamic model corrected by using open-loop test

corrected to seek the characteristics of the DHS and various simulations.

zone air temperature in Substation #3 (48 h) reaches 15 h similarly.

The purposes of doing OLT based on the developed dynamic model are stated hereby. Firstly, the mathematical model should be checked out with ideal condition to ensure the accuracy. Then, by applying the experience and operational data, the ideal dynamic model could be

Advanced Control Strategies with Simulations for a Typical District Heating System to Approaching Energy…

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The ideal conditions represent that outside and indoor air temperature and water mass flow rate in primary and secondary system are same as their design values. The affluent factors of both heat transfer area of each substation and terminal equal to 1. No solar radiation and internal heat gains exist in the ideal dynamic system. The heat losses from both water leakage

With these situations, the dynamic responses of the ideal model with the fuel control signal by 0.798 are shown in Figure 3. In addition to the zone air temperature in Substation #3, which is equal to 17.9C due to the huge thermal capacity of the floor heating structure, the supply and return temperatures from the heat source and substations are identical to the design conditions. Steady-state time of the water temperatures and zone air temperatures except for the

(a) (b)

(c) (d)

Figure 3. Dynamic responses of ideal model (a) Time(h), (b) Time(h), (c) Time(h), (d) Time(h).

4.1. The purposes of OLT

4.2. Ideal model of the DHS

and pipe network are ignored.

#### 3.3.4. Indoor air model

$$\mathcal{C}\_{z1}\frac{dT\_{z1}}{dt} = \mathcal{c}\_{w}(\mu\_{21}\mathcal{G}\_{21d} - 0.5\mathcal{G}\_{mk21})(T\_{s21z} - T\_{r21}) + \mathcal{q}\_{sls}F\_{s1} + \mathcal{q}\_{int}F\_1 - \mathcal{U}\_{m1}(T\_{z1} - T\_o) \tag{12}$$

$$\mathcal{L}\_{z2}\frac{dT\_{z2}}{dt} = c\_w(\mu\_{22}G\_{22d} - 0.5G\_{ml22})(T\_{s22z} - T\_{r22}) + q\_{sds}F\_{s2} + q\_{int}F\_2 - \mathcal{U}\_{cn2}(T\_{z2} - T\_o) \tag{13}$$

$$\left(\mathbb{C}\_{z3} + \mathbb{C}\_{\varepsilon}\right)\frac{dT\_{z3}}{dt} = \sigma\_w(\mu\_{23}\mathbf{G}\_{23d} - 0.5\mathbf{G}\_{\text{nl23}})(T\_{s23z} - T\_{r23}) + \eta\_{\text{sds}}F\_{\text{s3}} + \eta\_{\text{int}}F\_3 - \mathcal{U}\_{\text{en3}}(T\_{z3} - T\_o) \tag{14}$$

Zone air temperature dynamic responses can be represented in Eqs. (12)–(14). The net heat stored is related to the heat obtained from the circulation water in the secondary system, the solar radiation from south side windows, the internal heat gains and the heat transferred to the outside environment. Note that the thermal capacity in the terminal of floor heating is considered by accumulating the influence of the concrete structure.

#### 3.3.5. Pipe segment in the primary and secondary systems

$$\mathcal{L}\_{\text{segj}} \frac{dT\_{\text{segoutj}}}{dt} = c\_w G\_{\text{seginj}} T\_{\text{seginj}} - c\_w G\_{\text{segoutj}} T\_{\text{segoutj}} - Q\_{m \text{ksegj}} - Q\_{h \text{ksegj}} \tag{15}$$

The schematic diagram of a pipe segment is shown in Figure 2. The makeup water and the heat loss from the pipe insulation are considered to gather the water temperature left from the pipe segment. The supply water temperature from the pipe segment is related to the heat loss from the pipe segment, while the return water temperature has been considered in the heat losses from pipe insulation and makeup water. In Eq. (15), the net heat stored in the pipe segment equals to the heat received from the entrance minus the heat outlet from the exit and the heat losses from both makeup water leakage and pipe segment. Note that supply pipe segments do not consider the water leakage by the assumption. Letter j represents each pipe segment.

In summary, 29 dynamic equations are used to address the overall DHS mathematical model. The developed model is utilized to obtain system properties, simulate various dynamic responses of control strategies and compare with system energy consumption.

Figure 2. Schematic diagram of a pipe segment.
