Control of transients in water distribution networks by H<FONT FACE=Symbol>¥</FONT> Control

Abstract

In this work, a controller for regulating the transients in water distribution networks is established. The control technique is the H<FONT FACE=Symbol>¥</FONT> Control. The developed controller is applied to a water distribution network and the results of this application demonstrate that the technique allowed the establishment of a robust controller, capable of attenuating the disturbances in a suitable way, being effective in controlling the oscillations of the state variables in question.

Hydraulics transients; water distribution networks; H-infinity control

Control of transients in water distribution networks by H_¥ Control

M. H. Terra^I; J. A. T. dos Reis^II; F. H. Chaudhry^III; R. Schiavetto de Souza^IV

^IDepartment of Electric Engineering – EESC/USP Av. Trabalhador Sãocarlense, 400, Centro 13566-590 São Carlos, SP. Brazil [e-mail] ^IIFederal Center for Technical Education of Espírito Santo - CEFETES Av. Vitória, 1729,Jucutuquara 29040-333 Vitória, ES. Brazil [e-mail] ^IIIDepartment of Hydraulic and Sanitary Engineering EESC/USP [e-mail] ^IVDepartment of Hydraulics/ Transportation - UFMS 79070-900 Campo Grande, MS. Brazil [e-mail]

ABSTRACT

In this work, a controller for regulating the transients in water distribution networks is established. The control technique is the H_¥ Control. The developed controller is applied to a water distribution network and the results of this application demonstrate that the technique allowed the establishment of a robust controller, capable of attenuating the disturbances in a suitable way, being effective in controlling the oscillations of the state variables in question.

Keywords: Hydraulics transients, water distribution networks, H-infinity control

Introduction

With the increase in the levels of urbanization and consequent growth in the demands for water, the water distribution systems have become more encompassing and their operation more complex. At the same time, the lack of sources that offer water in suitable quantity and quality has forced a search for alternatives that allow better management of the available water resources.

In the last few years, several attempts have been carried made to establish control systems applicable to water distribution networks (Miyaoka & Funabash, 1984; Ormsbee & Lansey, 1994; Souza & Chaudhry, 1999; Cembrano, et al., 2000). The water distribution networks control systems have been implemented seeking to fulfill different objectives, amongst which hydraulic performance and economic effectiveness stand out. The hydraulic performance measures include the pressure levels, water quality and the determination of system reliability parameters; the economic effectiveness, in turn, is defined by distribution system operation and maintenance costs. The H_¥" Control (Doyle et al., 1989; Safonov et al., 1989; Doyle et al., 1992; Zhou et al., 1995; Terra et al., 2000) constitutes a technique that allows the development of robust automatic controllers, capable of assuring the fulfillment of criteria of hydraulic and economic effectiveness in water distribution systems, attenuating disturbances between input and output.

In this work, the problem of transient control in water distribution systems is discussed. A rigid model, used frequently for the evaluation of network transients, is presented and is used as a basis for the development of a H_¥" controller. After a brief presentation of H_¥" controllers design, a controller is established and applied to a hydraulic network.

This article is organized in the following way: item 2 discusses the problem of water distribution system transients, showing the hydraulic model and its linearization; item 3 presents the H_¥ Control technique; an example of H_¥ controller application to a water distribution network is presented in item 4, where the example network features are presented and the results discussed.

Nomenclature

a_t = cross-sectional area, m²

Ai = incidence matrixA = plant matrix of a finite-dimensional linear differential system

A= plant matrix of a fineite-dimensional linear differential system

Â_¥= auxiliary matrix

B = input matrix of a finite-dimensional linear differential system

C = output matrix of a finite-dimensional linear differential system

C_d = auxiliary matrix

d = diameter, m

E_I = auxiliary matrix

f = friction factor, dimensionless

f_v = valve head-loss coefficient, dimensionless

f(q) = vector of head- losses

F = auxiliary matrix

F_¥ = auxiliary matrix

g = acceleration of the gravity, m/s 2h_I = vector of independent heads

h_i = head vector

H_¥= Hamiltonian matrix

I = identity matrix

q_I = vector of independent pipe discharges

K = compensator

L_I , L_¥, M, Q_T = auxiliary matrices

Q_i = water demand vector

R = auxiliary matrix

t = time, s

u = vector of input perturbation

U = input vector

U_f = vector of the final equilibrium input

x = vector of state perturbations

X = vector of the state variables

X_f = vector of final equilibrium state variables

X_¥ = Riccati matrix

w = vector of input perturbation

W_I ,W_D = auxiliary matrices

y = vector of output signals

Y_¥= Riccati matrix

z = vector of output signals

Z _¥ = auxiliary matrix

Greek Symbols

s _max = maximum singular value

g = upper limit to the closed loop gain

r = spectral radius

l = eigenvalue

Hydraulic Transients in Water Distribution Networks

Transient flow in a network occurs when it changes from one permanent state to another. A flow change in a conduit system occurs due to a change or variation in the boundary conditions. Amongst the several boundary conditions that can introduce transients, the most usual and that require analysis frequently are:

• Starting or stopping the pumps;

• Variations in the demands or consumptions;

• Accidental or programmed changes in the valves regulations;

• Change in a reservoir rise.

Two different models have been commonly used for the analysis of transient flows in distribution networks: the rigid column models that do not take into account the water and conduit elastic effects and the models of elastic column, that consider such effects. The rigid models are usually employed for the description of the mass oscillation phenomena (Onizuka, 1986; Shimada, 1989); the elastic models, in turn, are used frequently for the analysis of water hammer studies (Chaudhry, 1979;Wylie and Streeter,1978). In this work, a rigid column model will be used following the proposed analysis.

Transients in Rigid Conduits

Transient flow in a closed conduit is described by nonlinear partial differential equations corresponding to conservation of mass and linear momentum (Chaudhry, 1979). The application of optimal control method adopted here requires, however, that these equations be formulated or transformed into linear ordinary differential equations. For the case of transients in rigid conduits, the momentum equation for a single pipe of length l reduces to the following ordinary differential equation:

where L' = l / g and the F_w = f l / (2 g d a_t²). In this equation, g is the acceleration of gravity, a_t the cross-sectional area of the conduit, t the time, f the friction factor and d the diameter of the conduit. The continuity equation (1) and the equations that establish the boundary conditions describe the hydraulic transients according to the rigid column model. Three types of boundary conditions are expected in hydraulic problems: nodes denoting surge tanks with free surface levels that will be defined in the following by [1]; nodes with a known flow vector and unknown heads denoted by [2]; and finally the nodes having unknown water consumptions but specified heads denoted by [3].

Network Equations

For a network consisting of different kinds of elements and specifications, the following system of state-space was proposed by Shimada (1992) for the vectors of independent pipe discharges (q_I) and heads (h_I) :

where

and

* denotes complex conjugate transpose matrix, f(q) is the vector of head-losses whose j-th element is given by

F_j q_j |q_j|

where

f_j is the friction factor, f_vjis the valve head-loss coefficient, d_j is the diameter and a_tj is cross-sectional area of the j-th pipe, L₁ is a diagonal matrix whose j-th element L_1,jj = l_j / (g a_tj), C_d is the diagonal matrix whose i-th diagonal element C_d,ii corresponds to the water elevation of variable level reservoir at the i-th node,

I is the identity matrix, is a linear transformation matrix, q_I is the vector of independent discharges, and, finally, A₁, A₂, A₃, h₁, h₂, h₃, Q₁, Q₂, Q₃are respectively the incidence matrices, head and water demand vectors for nodes [1] with unknown head and demand, nodes with known demands but heads unknown [2] and nodes with heads known but demands unknown [3].

State-Space Equations and Linearizations

The nonlinear equations (02) and (03) of the continuous system in the state-space form can be arranged as:

where f is a nonlinear vector function, U is the input vector as the control mechanism and X the vector of the state variables expressed as:

In order to analyze the transient states near the final equilibrium state, the system in Eq.(4) is linearized as:

with the following definitions for the perturbed state and control variables:

Here, x is the vector of state perturbations, u is the vector of input perturbations. The final equilibrium states X_f and U_f satisfy the following relationships:

Note that the constraints x < X_f and u < U_f must be met, where . denotes the norm of a vector.

H_¥ Control Design

H_¥ is a controller design method based on optimization, which guarantees stability and controlled system performance robustness. The purpose of this section is to describe the design procedure of an H_¥ controller (Chiang & Safonov ,1992; Doyle et al.,1992; Doyle et al., 1989; Francis, 1987; Maciejowski, 1989; Safonov et al., 1989; Zhou et al., 1995; Terra et al.,2000).

The H_¥ norm of a multivariable transfer function G is given by:

where sup denotes the supremum of s _max(.) and s _max denotes the maximum singular value. The H_¥ norm can be used to "shape" or to "restrict" transfer functions.

The Generalized Regulator Problem

Many control problems can be viewed as H¥ control problems. In practice, one wishes to find a controller that satisfies simultaneously some objectives, such as disturbance rejection and sensor noise attenuation. Consider a linear system, described by the transfer function P (see Fig.1).

System P has two sets of input signals, w and u. The vector u contains all control inputs which can be manipulated in such a way that the objectives are reached. The vector w contains all other inputs, such as reference signals, disturbances, and sensor noises. System P has two sets of output signals, z and y. The vector z contains all other variables that one wishes to control. Mathematically we have:

The generalized regulator problem consists of designing a linear system K such that, with u = K y, the closed loop system (Fig. 1) from w to z, has the following properties:

1. The closed loop is internally stable;

2. The H¥ norm of the closed loop is less than or equal to 1, i.e., ½½Á(P,K)½½_¥ < 1, where:

H_¥ Control Solution

The solution described here is a result obtained by Doyle et al. (1989). This solution makes use of two algebraic Riccati equations and designs a controller with the same number of states of the plant.

Given a block system P(2x2) and an upper limit g to the closed loop gain, the solution returns a compensator K(s) that stabilizes the system such that ½½Á(P,K)½½_¥ <g.

The following result is based on certain constraints on matrix D of the transfer function matrix of the plant P. These constraints may be ignored when applying the results of Safonov et al. (1989). The transfer matrix P is of the form

which represents the following set of equations:

The following assumptions are made:

1.

(A,B₁)

and

(A,B₂)

are stabilizable;

2.

(C₁,A)

and

(C₂,A)

are detectable;

3.

4.

The H_¥ solution involves two Hamiltonian matrices, given by (18) and (19):

Theorem 1: There exists an admissible controller such that ½½T_zw½½_¥ < g if and only if the following three conditions hold:

1. H¥ Î dom(Ric) and X_¥ := Ric(H_¥ ) > 0;

2. J_¥ Î dom(Ric) and Y_¥ := Ric(J_¥ ) > 0;

3. r (X_¥Y_¥ ) < g ²

where is the spectral radius of X_¥ Y_¥, l_i(X_¥Y_¥) is the i-th eigenvalue of X_¥Y_¥ and the notation X = Ric(H) is used to denote a solution for the algebraic Riccati equation that makes E-RX stable (all of its eigenvalues are on the left half plane). The algebraic Riccati equations associated with Hamiltonian matrices (18) e (19) are defined by:

The process of finding a sub-optimal H_¥ controller is iterative. Beginning with an arbitrary value of g , e. g., g = 1, if one of the conditions above fails, then g is too small and a solution does not exist; therefore, g should be increased.

Moreover, when these conditions hold, one such controller is given by

where , and

Results and Discussion

This section presents the distribution network in which the H_¥" control was applied, the established controller and some of the results obtained from the application of this controller to the distribution network.

Distribution Network Under Study

The simple network used by Shimada (1992) is chosen for implementing the H_¥" control formulations presented in this paper. The formulation is quite general, however, for the application to large size networks. The example network (Fig. 2) has 5 pipes, 2 variables level reservoirs with overflow weirs (nodes 1 and 2), 2 demands nodes (3 and 4), 2 constant level reservoirs (nodes 5 and 6) and 4 pressure reducing valves in pipes 1, 3, 4 and 5.

The pipe and node data are presented in Tables 1, 2 and 3. Tables 4 and 5 present the initial and final states of the network.

The H Control

For the H controller design, the matrices of state A, B, C and D were defined as follows:

The five selected state variables represent the deviations, in relation to the nominal conditions, of flows in conduits 2, 4 and 5 (Fig. 2) and of energies at nodes 1 and 2. To minimize these deviations, the system is controlled by closing or opening the pressure-reducing valves. The disturbances, in turn, are caused by the changes in demand outflows at nodes 3 and 4. In this design example, the state observer is not calculated.

The application of H_¥" Control leads to the dynamic controller solution (17), with the following solution to the algebraic equation of Riccati:

with g = 0.57, Y_¥" = [0] and L_¥" = [0].

These different disturbances are graphically represented in Fig. 3

Response of Controlled Network

The performance of the designed controller was evaluated through the analysis of temporal behavior of the controlled system state variables. The gain matrix presented in the last section, was used in the simulations of the system behavior for different external disturbances. In the simulations carried out, it was assumed that all the states were measured and that the system was initially in equilibrium (tables 4 and 5).

The external disturbances have been caused by variation in the water consumption presented in the distribution network nodes 3 and 4. Three different patterns of consumption variation have been considered:

1. A step function of 0.03 m³/s at the consumption nodes;

2. A step function of 0.06 m³/s at the consumption nodes;

3. An outflow variation at the consumption nodes in the form of a damped sine wave;

4. An outflow variation at the consumption nodes according to the daily variation pattern water consumption established after a proposed hydrogram by Yassuda and Nogami (1987).

The behavior of state variables x₁, x₂ and x₃, which correspond to the flow variations in relation to the equilibrium conditions in pipes 2, 4 and 5 of the network being considered, are graphically represented in Fig. 4 for each of the assumed disturbances in the outputs.

An examination of the graphs presented in Fig. 4 shows the disturbance evolution in the different state variables. One can observe that, after an initial period with large disturbances around the equilibrium condition, the designed controller leads the state variables to their nominal values (tables 4 and 5), regardless of the pattern of disturbances acting on the system. In all the simulations, the equilibrium condition was reached after a period of approximately 10 minutes (in all the graphs the convergence time is of approximately 600s). It is important observe that the H_¥ Control aims at attenuating disturbances with limited L₂-norm (). This is the case of the third disturbance, in which H_¥ Control shows the best performance. For instance, in the other two disturbance classes (step function and the hydrogram proposed by Yassuda and Nogami) the H¥ Control attenuates the disturbances but not with the same performance.

Additionally, it is worth noting that the flow behavior in pipes 2, 4 and 5 of the network (respectively variables x₁, x₂ e x₃) is different, for all the scenarios used in the simulations; in all the cases, the removal of water through the consumption nodes initially causes a decrease in the pipe 2 discharge and a large increase in the discharges in pipes 4 and 5 of the network. Further along time, however, this behavior periodically alternates itself until the equilibrium condition is reached.

Conclusions

This work demonstrated the application of the H¥ Control technique to control transients in water distributions networks submitted to external disturbances. In the simulations carried out, the disturbances represented sudden removal of water at different points of the distribution network and followed different temporal variation patterns.

The results showed that the computed controller is effective in controlling the discharge oscillations in different parts of the network, being capable of suitably rejecting the disturbances and of quickly conducting the system back to the desired condition.

It should be observed that the H¥ computational procedure for identifying the controller applied here to a simple hydraulic network can be usefully applied to larger networks once the rigid column conditions and those imposed by linearization are met with.

Article received December, 2001

Technical Editor: Atila P. Silva Frire

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Publication Dates

• Publication in this collection
  08 Sept 2003
• Date of issue
  Nov 2002

History

• Accepted
  Dec 2001
• Received
  Dec 2001