VALVE CONTROL OF PRESSURE SURGES IN HYDRAULIC SYSTEMS

 

B. PETRY ¹, R. K. GUPTA ²

 

¹ Professor of Hydraulic Engineering, ² Research Fellow

International Institute for Infrastructural, Hydraulic and Environmental Engineering

Westvest 7, 2624 AX Delft, The Netherlands.

Phone : 0031-15-2151715 ; Fax : 0031-15-2122921 ; E-mail : rkg@ihe.nl

 

 

ABSTRACT

The problem of analysis and control of surges in a pipe network is of considerable significance and finds many applications in engineering practice. Analysis of a system for prescribed transient behaviour of some components and design of other components to meet specifications can be termed transient synthesis. Mathematical formulation of such problems in a comprehensive and physically meaningful way is often neither apparent nor straightforward. The problem of solvability of resulting models is dependent on the manner in which design parameters and corresponding boundary specifications are distributed over the network. In this paper, a general treatment is given and the formulation is presented for an arbitrary network. Formulation of the problem is based on graph-theoretic concepts. Possible boundary conditions are considered by categorising these into five element types. Necessary and sufficient conditions for the existence and uniqueness of a solution, which are important for an effective modelling of these problems, are also stated. Among other advantages, the algorithm provides an automatic separation of dependent and independent variables and results in a minimum set of independent variables for the numerical solution of the problem. Moreover, valve operation rules are calculated explicitly as valve openings are treated as dependent variables.

 

The principle behind the methodology developed in this paper is that transient behaviour in some components of the system is specified and some other components are kept undetermined and to be designed in order to meet the imposed specifications. Analysis of the system directly yields the transient behaviour of these components, which are then accordingly designed. Application of this methodology is illustrated by an example. Valve operations are determined from direct analysis to satisfy imposed specifications. In general, the algorithm is found to be useful for transient design of other components of hydraulic networks such as surge tanks, conveyance elements.

 

Keywords: Pressure surges, Analysis, Control, Pipe network, Transient design, Slow transients, Optimal valve operation.

 

INTRODUCTION

Problems of analysis and control of slow transients in complex pipe networks can be divided into two types. The first is the analysis problem in which hydraulic performance of the system is to be determined for specified valve operations. No constraints on head and/or flow rates at the valves or in the pipes are imposed and, thus, the time integration of a system of ordinary differential equations can be made in a straightforward manner by using an adequate numerical technique. The second is the analysis-control problem. In this problem, valve operations are treated as variables and are determined in such a way as to transfer the system from an initial steady-state to a final steady-state condition under specified transient behaviour in some components of the network. Resulting valve operations extend over the entire duration of transients. Design of valve operations can further be achieved by optimising a selected objective function.

 

CONCEPTUAL BACKGROUND / PROBLEM SOLVABILITY

The present method for the analysis and control of slow transients is based on graph-theoretic concepts. All system components of a network are either elements or nodes of a graph. In a network, a node is characterised by a particular head, H(t) and an outflow discharge, Q(t). Similarly, an element is associated with a particular head loss, h(t) and a discharge, q(t). Depending upon whether the head/headloss and/or discharge vector is a known quantity, boundary conditions can be categorised into five categories called, hereinafter, element types. These five element types are given in Table 1.

 

Table 1. Element Types

 

For the development of solvability conditions and the mathematical formulation of the problem, a extended network is constructed by considering one of the type 1 node as a reference node and by joining all outflow nodes to the reference node. In the extended network, all the type 5 elements can be divided in two categories called, hereinafter, element type a and element type b. Type a elements are always the branches of a spanning tree and type b elements are always the chords. Figure 1(b) shows the extended network for the network shown in Fig. 1(a).

 

Fig. 1 - (a) Hydraulic Network ; (b) Network Graph (closed)

 

In principle, the problem is mathematically well posed if the number of unknown variables equals the number of boundary specifications imposed. Thus, it is evident that for a well posed problem the number of type 3 elements must be equal to the number of type 4 elements. Moreover, the solvability of these problems is dependent on the manner in which these elements are distributed over the network. In other words, boundary specifications in terms of flow/head variations and the design parameters can not be distributed arbitrarily in a network but should obey certain rules for the existence and uniqueness of a solution. For a network consisting of five types of elements as described earlier, a necessary and sufficient condition for the existence and uniqueness of a solution is defined by the following theorem.

Theorem

In the extended network consisting of elements type 1,2,3,4 and 5, a necessary and sufficient condition for the existence and uniqueness of a solution is the existence of at least one tree T_1,3,a and at least one T_1,4,a having the following properties:

1.      T_1,3,a is a spanning tree. Its branches include all elements type 1 & 3 and a subset of the elements type 5 (called "a"). The co-tree of T_1,3,a includes all elements type 2 & 4 and the remaining elements type 5 (called "b").

2.      T_1,4,a is a spanning tree. Its branches include all elements type 1 & 4 and the same subset of the elements type 5 (called "a") as included in spanning tree T_1,3,a. The co-tree of T_1,4,a includes all elements type 2 & 3 and the remaining elements type 5 (called "b").

A demonstration of this theorem is not provided here due to limitation of space.

 

MATHEMATICAL FORMULATION

In the present method of analysis, continuity equations have been derived considering spanning tree T_1,4,a and the loop head loss equations have been formulated for the spanning tree T_1,3,a. This formulation automatically separates the dependent and independent variables and calculates the head and discharge variables in system components with unknown parameters explicitly. Continuity equations can be expressed as flow in tree elements of the spanning tree T_1,4,a created by chords. These equations are as follows:

(1)

(2)

(3)

Note that the only independent variables are flows in type b elements. Loop head loss equations for the spanning tree T_1,3,a can be expressed as

(4)

(5)

(6)

L and L' are loop incidence matrices for spanning trees T_1,3,a and T_1,4,a. Surge equations for all the pipes and surge tanks i.e. element type a and b are given by

(7)

(8)

where l, D, A and f are length, diameter, area and Darcy-Weissbach friction of the pipe respectively and A_s represents the surge tank area. In addition, equation for discharge through the valves is given by

(9)

where subscript 0 denotes initial steady-state and J is the dimensionless valve coefficient with respect to initial steady-state condition. Inserting eq. 3 in eq. 7 for h_a and then using the resulting equation along with eq. 7 for h_b in eq. 6 gives a set of ordinary differential equations in which the only variable is q_b. A system can be analysed by solving this set of equations for q_b and eq. 8 for H_s using a numerical scheme such as the Runge-Kutta fourth order. Once q_b and H_s are known, all other variables are determined directly from eqs. 1-7.

 

GENERAL SOLUTION PROCEDURE AND APPROACH TO SYSTEM COMPONENT DESIGN

General solution procedure and approach to system component design is outlined in Figure 2. First analysis of the system provides flow and head loss variables in each component. Operation rules, J(t), for the valves considered as type 4 elements are determined from eq. 9 after calculating q₄ and h₄ at each time step. Valve operations obtained from this analysis extends until final steady-state is reached. Methodology is applicable for real time control of slow transients. However, in many applications such a long valve manoeuvring is undesirable and, hence, design of valve operation rules is made for a specified time limit by optimising the calculated valve operation rules obtained from the first analysis. After determining the valve operation rules system can be transferred to the final steady-state by considering valves as type 5 elements. Note that in principle any system component can be considered as type 4 element. Result from the first analysis provides the transient behaviour of this component which can be optimised to match a desired component behaviour.

 

 

Fig. 2 - Methodology for System Analysis and System Component Design

 

APPLICATION TO DESIGN OF VALVE OPERATIONS

A valve with unknown coefficient is considered as type 4 element. Network analysis yields discharge, q₄, and head loss, h₄ and, thus, valve openings are determined directly using eq. 9. Let this openings be called J₄^c. If the valve operations are desired to be completed at the time t=T, the actual valve openings may be represented by a Fourier series as follows:

(10)

where J₀ and J_f are valve openings corresponding to initial and final steady-state. In order to achieve that J₄ is as close as possible to J₄^c, a minimisation problem is formulated with the objective function being

(11)

Parameters a_n are determined using the condition ¶f/¶a_n=0. Knowing the values of J₄^c, J₄ is determined using eqs.10-11 for the specified time of valve operations T. With calculated J₄ and specified J₃, the system is subjected to a final analysis.

 

EXAMPLE

Network shown in Fig. 1(a) is taken as an example for which the system parameters are given in Table 2. Using the present methodology the system is transferred from an initial steady-state to a final steady-state as given in Table 3.

 

Table 2. Characteristics for Pipe Network in Fig. 1(a)

 

Table 3. Initial & Final Steady-States : Discharges, Heads and Valve Openings

 

Valve 1 and 3 have been taken as type 3 element in which a transient behaviour in terms of linear increase of head loss from initial steady-state value to final steady-state value in 30 seconds and linear valve closure, J₃=1 to .15 for valve 1 and J₃=1 to .0.2 for valve 3, in 30 seconds are specified. System is analysed and resulting J₄^c, flow and head variations in pipes are shown in Fig. 3,4and 5 respectively. J₄^c are optimised to get valve operation rules, J₄, for the specified operation time of 30 seconds. Resulting J₄ are shown in Fig. 3. System is again analysed using optimised J₄ and specified J₃. Figs. 4 and 5 also show the resulting flow and head variations in pipes. System is smoothly transferred to final steady-state without violating much the specified transient behaviour in element type 3 i.e. valve1 and 3.

 

Fig. 3 - Valve Operation Rules

 

Fig. 4 - Discharge Variation in Pipes and Valves

 

Fig. 5 - Nodal Head Variations

 

CONCLUSIONS

The problem of analysis and control of slow transients under specified transient behaviour in some part of the system can be termed as transient synthesis. The mathematical formulation of the problem is not straightforward as problem solvability is dependent on topological allocation of design variables and boundary specifications. In this paper, the problem has been mathematically formulated for a arbitrary network and conditions for solvability of the problem have been developed which are important for an effective modelling of these problems. Present algorithm, based on loop formulation, is more efficient as it provides a minimum set of state variables with automatic separation of dependent and independent variables. Moreover, the valve operation rules are calculated explicitly. The proposed methodology can be used for transient system component design. Design of valve operation rules have been carried out for a selected example. Other system components such as surge tank, pipe, air vessels and so on can also be designed using the developed methodology.

 

REFERENCES

1.      Onizuka, K., "System dynamics approach to pipe network analysis." J. Hydr. Engg., ASCE, 112(8), 1986.

2.      Petry, B. and Gupta, R. K., "An improved approach to surge tank design in high head hydropower systems." Conference on Modelling, Testing and Monitoring of Hydropower Plants-III, France, 1998.

3.      Shimada, M., "Graph theoretical model of slow transient analysis." J. Hydr. Engg., ASCE, 115(9), 1989.

4.      Shimada, M., "State space analysis and control of slow transients in pipes." J. Hydr. Engg., ASCE, 118(9), 1992.

5.      Streeter, V.L. and Wylie, E. B., Fluid Transients in Systems, Prentice Hall, NJ., 1993.