Graphs are not only suitable for describing relationships among the input, the state, and the output variables of dynamic component models, but also suitable for describing link relationships and informationflow among parts of a complex sys- tem. It helps us have an overall understanding on the complex system and analyze interactions among the system’s components.
According to the definition of graphs [1], a graph is made up of branches of nodes and arcs connecting the nodes. Arcs are also known as branch lines, straight lines, or edges. Taking the graph in Fig.4.11, for example, it consists of nodes X1X7 and arcs L1L7 which can be expressed by Eq. (4.17)
Gẳ ðX;Lị ð4:17ị
where
Xẳ fX1;X2;. . .;Xng; Lẳ fL1;L2;. . .;Lng:
In the graph representation for a system, the graph nodes denote component units of the system, and the directed arcs for mass and energyflows among unit components. The directed graph can offer us an overall structure information of a system and assist us to develop the system model.
In order to facilitate computer recognition of the system structure, the directed graph is often rewritten into a matrix form also known as structural matrix. Suppose a graph is composed ofm nodes, then the system structural matrix is defined as below:
Sẳ ẵsi;jmm iẳ1;2;. . .;m;jẳ1;2;. . .;m ð4:18ị Ifsi;jẳdi;j, it means there exists a directed arcflow from nodeXitoXj, and the flow dimension isdi;j; and ifsi;jẳ0, it means there is no arc connection between nodeXiandXj.
Fig. 4.11 Graph
representation for a system
Assuming that each arc in the system (Fig.4.11) is of oneflow dimension, the structure matrix of the system can be written as below:
Sẳ
X1 X2 X3 X4 X5 X6 X7 X8 X1
X2
X3
X4
X5
X6
X7
X8
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
2 66 66 66 66 66 4
3 77 77 77 77 77 5
ð4:19ị
From the above structure matrix, we can get the structural characteristics of the system as below:
(1) For a system consisting of m nodes, its structural matrix consists of at least (m− 1) nonzero elements (the simple mode of connection between the nodes is series connection);
(2) If all the elements in theith column are zero (0), thejth row must contain at least one nonzero element and vice versa;
(3) The nodes corresponding to the rows or columns whose elements are all zero represent a unit known as system terminal unit. The all-zero row corresponds to output terminal unit, and the all-zero column corresponds to the input terminal unit;
(4) Series unit is represented by the nonzero elements above the main diagonal of the matrix;
(5) The parallel output terminal is represented by multiple nonzero elements in a row, and the nonzero elements equal to the number of parallel branches.
Likewise, the parallel input terminal is represented by multiple nonzero ele- ments in a column;
(6) The nonzero elements below the main diagonal of the structure matrix rep- resent the system feedback.
To sum up, a system structure matrix can clearly illustrate the position of components and their mutual relationship in the system. Meanwhile, it can reflect the system hierarchy as well.
4.2.2 Case Study
Taking the HVAC system in Fig.2.40, for example, the graphic description for the system can be obtained according to the graph-theory modeling approach, as shown in Fig.4.12, where ‘Srefri’ stands for refrigeration system, ‘Sconw’ for cooling coolant system which consists of cooling coolant pump, cooling tower, and pipes,
‘Schw’for chilled coolant system which consists of chilled coolant pump and pipes,
‘Sahu’for air-handling system which consists of surface heat exchanger and supply air fan, ‘Sair,s’ for supply air system, ‘Sair,er’ for exhaust-return air system, and
‘Sroom’for air-conditioned room.
Based on the directed graph in Fig.4.12, the HAVC system can be expressed by structural matrix as below:
Sẳ
Sconw Srefri Schw Sahu Sair;er Sair;s Sroom;1 Sroom;2 Senv Sconw
Srefri Schw Sahu Sair;er
Sair;s Sroom;1 Sroom;2 Senv
0 2 0 0 0 0 0 0 0
2 0 2 0 0 0 0 0 0
0 2 0 2 0 0 0 0 0
0 0 2 0 0 3 0 0 0
0 0 0 3 0 0 0 0 3
0 0 0 0 0 0 3 3 0
0 0 0 0 3 0 0 0 0
0 0 0 0 3 0 0 0 0
3 0 0 3 0 0 4 4 0
2 66 66 66 66 66 66 4
3 77 77 77 77 77 77 5
ð4:20ị In Eq. (4.20),S12 =S21= 2 represents from cooling coolant system (Sconw) to cold supply system (Srefri) and from cold supply system (Srefri) to cooling coolant system (Sconw), respectively, and both have two variables i.e., cooling coolantflow rate and cooling coolant temperature. S23 =S32 = 2 represents from cold supply system (Srefri) to chilled coolant system (Schw) and from chilled coolant system (Schw) to cold supply system(Srefri), respectively, and both have two variables i.e., chilled coolant flow rate and chilled coolant temperature. S44=S46= S54 =S59 =S67=S68=S75=S85=S91= S94= 3 represents from air-handling system (Sahu) to air supply system (Sair,s), from exhaust-return air system (Sair,er) to air-handling system (Sahu) and external environment (Senv), from air supply
Sroom,1
Sroom,2 Sconw
Senv
Srefri
Sair,er
Sair,s
Senv
Schw
Sahu
Senv
Fig. 4.12 Directed graph for the HVAC system presented in Fig.2.40
system (Sair,s) to conditioned room (Sroom), from air-conditioned room (Sroom) to ambient environment (Senv), and from ambient environment (Senv) to cooling coolant system (Sconw) and air-handling system (Sahu), respectively, and all have three variables i.e., airflow rate, air temperature, and air humidity; S97= S98= 4 represents from ambient environment (Senv) to air-conditioned room (Sroom), respectively, and both have four variables, i.e., air flow rate, air temperature, air humidity, and solar radiation.
Figure4.13 shows directed graph for subsystems in the HVAC system, where Xcoilstands for surface heat exchanger;Xfanfor fan;Xpumpfor pump (the subscripts
‘1’and ‘2’denote cooling coolant pump and chilled coolant pump, respectively);
Xtowerfor cooling tower;Xsductfor three-way duct (the subscripts‘1’and‘3’denote split-flow duct in exhaust-return air system and supply air system, respectively, and the subscripts‘2’and ‘4’denote confluent-flow duct in exhaust-return air system and supply air system, respectively.);Xzductfor straight-through duct (the subscripts
‘1’and‘2’denote duct from the air-handling unit toXsduct,1and that fromXsduct,1to air-conditioned room, respectively, and the subscript‘3’denotes duct fromXsduct,2
toXsduct,3);Xzpipe for straight-through pipe (the subscripts‘1’and ‘2’denote pipe from cooling tower to chiller’s condenser and that from chiller’s condenser to cooling tower, respectively, and the subscripts‘3’and‘4’denote pipe from chiller’s evaporator to surface heat exchanger and that from surface heat exchanger to chiller’s evaporator);Xspipefor three-way pipe;Xfafor air valve;Xfwfor pipe valve;
andXroom for air-conditioned room.
Sahu Sahu
Ambient
Directed graph for Sahu
Sair,er
Xcoil
Schw
Ambient
environment Xsduct,4 Directed graph for Sair,s Xzduct,1
Air-conditioned room Xfa,2
Xzduct,2 Xfa,1 Xsduct,1
Xfan Sair,s
Xzduct,3
Xsduct,3
Directed graph for Sair,er Xsduct,2
Um Outdoor
environment
Condenser of chiller
Directed graph for Schw
Directed graph for Sconw Xpump,1 Xzpipe,2
Xtower Xzpipe,1
Xzpipe,4 Xfw
Xpump,2 Xzpipe,3
Um
Um
Xcoil
Evaporator of chiller
Fig. 4.13 Directed graph for subsystems in the HVAC system