Property 6 : In a network, sum of elements type 2 and 4 can not be more than number of chords
6 Principles of Control of Pressure Surges
6.5 Applications of Determined Models
Determined models are useful in many different applications of network analysis and control of pressure surges. Problems of design of valve operations for transferring the network from an initial steady state to a final steady state with the specified transients in terms of head loss and/or discharge variations in some parts of the network are of immense practical use. Practical applications are found in the fields of transmission and distribution networks, industrial piping systems, hydropower systems, pumping stations etc. Design of valve operations for changing the outflow at different locations in the network in the specified trajectories, for changing the outflows with the specified head loss in the elements or head at nodes are few examples of such applications.
Application of partial control determined problem
As an example, consider a network as shown in Fig. 6.10. Network consists of one reservoir, one surge tank and four control valves. Characteristics of the network are given in Table 6.1. It is desired to transfer the network from an initial steady state to a final steady state with the specified transients at node B and D i.e. with specified outflow and head variations at these nodes.
1
Number along the edges shows pipe number
Fig. 6.10 A pipe network as an example
Table 6.1. Characteristics for Pipe Network in Fig. 6.4
Pipe 1 2 3 4 5 6 7 8 9 10 11
l (km) 1.0 1.0 1.5 1.0 1.0 1.5 .05 1.0 1.5 1.0 1.0 D (m) 2.8 0.95 2.45 1.2 1.5 1.3 1.0 0.85 1.0 1.3 1.25 f = 0.016 for all pipes ; Area of surge tank = 40 m2
Table 6.2. Initial & Final Steady-States : Discharges, Heads and Valve Openings
q (m3/s) Head (m) q (m3/s)
Valves 1 and 3 are taken as type 3 elements in which a transient behaviour in terms of linear increase of head loss from initial steady state value to final steady state in 30 in
30 seconds and a linear valve closure, τ = 1 to 0.15 for valve 1 and τ = 1 to 0.2 for valve 3 in 30 seconds are specified. Initial and final steady states of the network are given in Table 6.2.
Valves 2 and 4 are modelled as type 4 elements. All other elements except reservoir of the network are type 5 elements.
The resulting set of ordinary differential equations is numerically integrated using Runge Kutta fourth order method. A time step of 1 second is selected. Analysis is carried out for 300 seconds.
The resulting valve operation rules for valves 2 and 4 are shown in Fig. 6.11. Figs.
6.12 and 6.13 show discharge variations through valves and few pipes and nodal head variations respectively.
0 0.2 0.4 0.6 0.8 1 1.2
0 100 200 300
Time (seconds)
Dimensionless valve opening (tau)
V 2
V 4
Fig. 6.11 Valve operation rules
0 5 10 15 20 25
0 100 200 300
Time (seconds)
Discharge (m^3/s)
q3
qv2
qv4
q8
q8
Fig. 6.12 Discharge variations in pipes and valves
0 20 40 60 80
0 100 200 300
Time (seconds)
Head (m)
HE
HD
HC
HA
Fig. 6.13 Nodal head variations
Application of full control determined problem
Consider network shown in Fig. 6.14 as an example. Network has 24 pipes, 2 reservoirs and 12 valves. Network elements data is given in Table 6.3. It is desired to change the flow in the network from an initial steady state to a final steady state, as given in Tables 6.3, 6.4 and 6.5, in 60 seconds with following specifications:
1. Flow through valves 2, 5, 6, 7, 10 and 11 is changed smoothly based on the following equation:
3 3 T 2 o
2 T
o o t
T ) q t 2(q
T ) q q 3(q
q = − − + − (6.39)
These elements are, thus, modelled as type 2 elements.
2. Head variation at node B is given as shown in Fig. 6. 18 with a maximum head rise limited to 62 m. Hence, node B is modelled as type 3 node.
3. Pipe elements 5, 7, 15 and 16 are considered as type 3 elements. Discharge variation through these pipes is changed smoothly as given in Fig. 6.15. These pipe elements are also modelled as type 3 elements.
Valves 1, 3, 4, 8 and 9 are modelled as type 4 elements. All other pipe elements are modelled as type 5 elements.
Network has control valves in its every loop. Hence, the system can be transferred from initial steady state to final steady state in the given time without any residual transients.
Analysis is carried out using the developed algorithm. Valve operation rules are shown in Figs. 6.16 and 6.17. Nodal head variations are shown in Figs. 6.18 and 6.19.
1 A 2 B
R1 3
V12
4 5 C 6 N
V1
V2 V3
D 7 8
E O
9 V4
V5
10 F 11 12
R2 G V7
Q
13 14
H 15 I 16 J 17 V6 P
18 19 20
V8 V9
K 21 22 M 24 V10 23 L S
R V11
Fig. 6.14 A pipe network as an example
0 1 2 3 4
0 10 20 30 40 50 60
Time (sec.) Discharge (m3 /s)
7 16
15
5
Fig. 6.15 Discharge variations in pipes as specifications
Table 6.3 Network pipe data and initial and final flows
Table 6.4 Initial and final valve coefficients Valve
Table 6.5 Initial and final heads at nodes Node Initial head
(m) Final head
0 1 2 3 4 5 6
0 10 20 30 40 50 60
Time (sec.)
τ
10
4
6 2
8
12
Fig. 6.17 Valve operation rules for valves 2, 4, 6, 8, 10 and 12
0 20 40 60 80
0 10 20 30 40 50 60
Time (sec.)
Head at node (m)
C A
B
D
P F
Fig. 6.18 Head variations at nodes
0 15 30 45 60 75 90
0 10 20 30 40 50 60
Time (sec.)
Head at node (m)
N
Q
H O
S
R
Fig. 6.19 Head variations at nodes
6.6 Concluding Remarks
In networks, control of flows is generally carried out through valve operations.
Operational flow control problems involve determination of valve operation rules to transfer a network from an initial steady state to a final steady state. In many practical problems, it is desired to have a specified variation of outflows/discharges and/or head/head loss in some parts of the network during the transient period. In these problems, conventional procedures are based on analysis approach i.e. use of trial and error methods for the design of valve operations. Development of procedures for the direct design of valve operations for the specified transients i.e. synthesis approach is essential for avoiding tedious trial and error procedures.
Design of valve operations for the specified transients can be termed as transient synthesis or transient design. Depending upon whether the transient specifications are equal or less than the control variables, two types of control problems exist, determined problem and underdetermined problem. This chapter deals with the determined problems.
In complex networks, the mathematical formulation of such problems in a comprehensive and meaningful form is often neither immediately apparent nor straightforward. The problem of solvability of resulting models is dependent on the network flow distribution and on the manner in which the decision parameters and corresponding boundary specifications are topologically allocated. In this chapter, problem has been mathematically formulated for an arbitrary network consisting of
possible transient specifications or boundary conditions. Determined models should obey necessary and sufficient conditions for the existence and uniqueness of a solution as derived in chapter 4.
The developed algorithm for the solution of determined models is more efficient as it provides a minimum set of ordinary differential equations with automatic separation of dependent and independent variables. Moreover, the valve operation rules are calculated explicitly.
Transfer of network from an initial steady state to a final steady state within specified time with no residual transients depends on the network controllability. For full controllability i.e. the case when no residual transients exist at the end of valve operations, each loop of the network should possess a control valve. If the network doses not possess valves in each loop, residual transients will persist.
From the example of a network, it is clear that in partial control problems though the transient specifications at desired locations are met while transferring the network from initial to final steady state, residual transients exist in the network. In such case, valve operates for a long time till the network attains final steady state. Such long operation of valves is generally undesirable. However, operation of valves can further be designed for the specified time. This problem has been discussed in chapter 8.